This course develops the variational theory of geometric objects that arise as critical points of area and energy. It begins with minimal surfaces, viewed as area-minimizing or stationary submanifolds, and then moves to harmonic maps, which play the analogous role for the Dirichlet energy between Riemannian manifolds. The central theme is how analytic tools reveal geometric structure: first variation identifies the Euler-Lagrange equations, while later chapters study stability, regularity, and the behavior of solutions under limiting processes. The chapters are arranged to move from foundational variational ideas to deeper analytic and geometric consequences. After introducing the first and second variations of area, the course studies minimal graphs, Bernstein-type results, monotonicity formulas, blow-up analysis, tangent cones, compactness, and curvature estimates, culminating in Plateau-type existence and regularity theory. The second half shifts to harmonic maps, starting with the tension field and Bochner identities, then addressing weak harmonic maps and regularity issues, and ending with existence via heat flow and the [Eells-Sampson theorem](/theorems/5704). Together, these topics show how nonlinear PDE methods organize the study of minimal and energy-critical geometry. # Introduction Geometric analysis studies geometric objects through variational problems and the elliptic equations that express their criticality. This course focuses on two model theories: minimal submanifolds, which are critical points of area, and harmonic maps, which are critical points of Dirichlet energy. The common theme is that geometry enters both the functional and the Euler--Lagrange equation, so analytic estimates must be read together with curvature, topology, and compactness. The course continues the foundational material on Riemannian geometry, elliptic PDE, Sobolev spaces, and the [calculus of variations](/page/Calculus%20of%20Variations), and prepares for later work on geometric flows and nonlinear regularity theory. ## The Two Variational Problems What does it mean for a geometric object to be the best representative in its class? For submanifolds, the quantity to minimise or make stationary is area. For maps between Riemannian manifolds, the quantity is energy, measuring how much the map stretches tangent vectors on average. The opening comparison is useful because the two theories share a variational structure while differing sharply in their compactness and singularity behaviour. [definition: Area Functional For An Immersion] Let $M^m$ be a smooth manifold, let $(N^n,h)$ be a Riemannian manifold, and let $M'\subset M$ be relatively compact. The area functional over $M'$ is the map $\operatorname{Area}(\cdot;M'):\operatorname{Imm}(M,N)\to \mathbb R$ defined by \begin{align*} \operatorname{Area}(F;M')=\int_{M'} d\operatorname{vol}_{F^*h}. \end{align*} where $\operatorname{Imm}(M,N)$ denotes the smooth immersions $F:M\to N$ for which $F^*h$ is the induced Riemannian metric on $M$. [/definition] The notation records that the area is not merely a property of the image set when self-intersections or parametrisations matter. To compare area with the second major functional in the course, we next put maps between fixed Riemannian manifolds into the same variational format. [definition: Dirichlet Energy Of A Map] Let $(M^m,g)$ and $(N^n,h)$ be Riemannian manifolds, and let $M'\subset M$ be relatively compact. The Dirichlet energy over $M'$ is the functional $E(\cdot;M'):C^\infty(M,N)\to \mathbb R$ defined by \begin{align*} E(u;M')=\frac12\int_{M'}|du|^2_{g,h}\,d\operatorname{vol}_g. \end{align*} [/definition] The area functional varies the geometry of a submanifold, while the energy functional varies a map defined on a fixed domain. Both functionals are local integrals depending on first derivatives, so their Euler--Lagrange equations are [second-order elliptic equations](/page/Second-Order%20Elliptic%20Equations) in favourable gauges or coordinates. [example: Flat Models] For an affine plane in $\mathbb R^3$, write its parametrisation as $F(x^1,x^2)=p+x^1a_1+x^2a_2$, where $a_1,a_2$ are linearly independent constant vectors. Then $\partial_iF=a_i$, so the induced metric has coefficients $g_{ij}=\langle a_i,a_j\rangle$, which are constant. Also $\partial_i\partial_jF=0$ for all $i,j$, hence the second fundamental form satisfies \begin{align*} A_{ij}=(\partial_i\partial_jF)^\perp=0. \end{align*} Taking the trace with the inverse induced metric gives \begin{align*} H=g^{ij}A_{ij}=0. \end{align*} By the [first variation formula](/theorems/2728) developed below, vanishing mean curvature is exactly the stationarity condition for area under compactly supported normal variations, so affine planes are stationary for area. For maps $u:\mathbb R^m\to\mathbb R^n$, write $u=(u^1,\ldots,u^n)$ and vary by $u_t=u+t\varphi$, where $\varphi\in C_c^\infty(U;\mathbb R^n)$ on a relatively compact [open set](/page/Open%20Set) $U\subset\mathbb R^m$. The Euclidean Dirichlet energy is \begin{align*} E(u_t;U)=\frac12\int_U\sum_{\alpha=1}^m\sum_{i=1}^n(\partial_\alpha u^i+t\partial_\alpha\varphi^i)^2\,dx. \end{align*} Differentiating the displayed polynomial in $t$ at $t=0$ gives \begin{align*} \frac{d}{dt}\Big|_{t=0}E(u_t;U)=\int_U\sum_{\alpha=1}^m\sum_{i=1}^n\partial_\alpha u^i\,\partial_\alpha\varphi^i\,dx. \end{align*} Integrating by parts in each coordinate, with no boundary term because $\varphi$ is compactly supported in $U$, gives \begin{align*} \int_U\sum_{\alpha=1}^m\sum_{i=1}^n\partial_\alpha u^i\,\partial_\alpha\varphi^i\,dx=-\int_U\sum_{i=1}^n\left(\sum_{\alpha=1}^m\partial_\alpha^2u^i\right)\varphi^i\,dx. \end{align*} Since $\Delta u=(\Delta u^1,\ldots,\Delta u^n)$ with $\Delta u^i=\sum_{\alpha=1}^m\partial_\alpha^2u^i$, this is \begin{align*} \frac{d}{dt}\Big|_{t=0}E(u_t;U)=-\int_U\langle \Delta u,\varphi\rangle\,dx. \end{align*} Stationarity for every compactly supported $\varphi$ is therefore equivalent to $\Delta u=0$ componentwise. Thus, in the flat models, the geometric equations reduce to $H=0$ for affine planes and to the ordinary harmonic equation for vector-valued functions. [/example] ## Stationarity As An Euler--Lagrange Equation How does a variational definition become a differential equation? The bridge is first variation: perturb the object by a compactly supported variation, differentiate the functional at time zero, and identify the coefficient of the variation field. The resulting equation is geometric because the permissible variations and the first derivative of the functional are expressed using the ambient or target Riemannian structure. [definition: Stationary Point Of A Geometric Functional] Let $\mathcal A$ be a class of smooth geometric objects and let $\mathcal F:\mathcal A\to \mathbb R$ be a functional. An object $X\in \mathcal A$ is stationary for $\mathcal F$ if for every compactly supported smooth variation $(X_t)_{|t|<\varepsilon}$ with $X_0=X$, one has \begin{align*} \frac{d}{dt}\Big|_{t=0}\mathcal F[X_t]=0. \end{align*} [/definition] Stationarity is weaker than minimisation, so the definition is not useful until one can compute the first derivative and recognize when it vanishes. The obstruction is that area and energy live on different geometric objects: an immersion changes by a vector field along its image, while a map into a target changes by a section of the pullback tangent bundle. The first variation formula identifies the quantity paired with these arbitrary compactly supported variations, giving mean curvature for area and the tension field for energy. [quotetheorem:5658] [citeproof:5658] This theorem is the conceptual entrance to the course because it turns variational language into geometric PDE. Compact support is the condition that removes boundary terms from the first variation; without it, stationarity would have to be supplemented by boundary conditions such as fixed boundary values, free-boundary orthogonality, or prescribed boundary motion. Normal variations are enough for area because tangential variations merely reparametrise the immersed submanifold in the interior, whereas for maps into $N$ the variation field must lie along $u$ and remain tangent to the target constraint. The theorem does not say that a stationary object minimises the functional, nor does it give regularity for weak critical points. A saddle point of area or energy can still satisfy the Euler--Lagrange equation, so the next questions are second variation, stability, and compactness. The early lectures make this first variation principle concrete for hypersurfaces and graphs; the later lectures repeat the same logic for maps, where the curvature of the target contributes extra lower-order terms. [example: Minimal Graphs And Harmonic Functions] Let $U\subset\mathbb R^m$ be open and let $u:U\to\mathbb R$ be smooth. Its graph is parametrised by $F(x)=(x,u(x))$, and its coordinate tangent vectors are \begin{align*} \partial_iF=e_i+(\partial_i u)e_{m+1}. \end{align*} Hence the induced metric has coefficients \begin{align*} g_{ij}=\langle \partial_iF,\partial_jF\rangle=\delta_{ij}+(\partial_i u)(\partial_j u). \end{align*} Equivalently, $(g_{ij})=I+\nabla u\otimes \nabla u$, so the rank-one determinant identity gives \begin{align*} \det(g_{ij})=1+|\nabla u|^2. \end{align*} Thus the graph area over a relatively compact open set $U'\Subset U$ is \begin{align*} \operatorname{Area}(u;U')=\int_{U'}\sqrt{1+|\nabla u|^2}\,dx. \end{align*} Now vary the graph vertically by $u_t=u+t\varphi$, where $\varphi\in C_c^\infty(U')$. Since $\nabla u_t=\nabla u+t\nabla\varphi$, the chain rule gives \begin{align*} \frac{d}{dt}\sqrt{1+|\nabla u_t|^2}=\frac{\langle \nabla u+t\nabla\varphi,\nabla\varphi\rangle}{\sqrt{1+|\nabla u_t|^2}}. \end{align*} Setting $t=0$ in this identity gives the first variation \begin{align*} \frac{d}{dt}\Big|_{t=0}\operatorname{Area}(u_t;U')=\int_{U'}\left\langle \frac{\nabla u}{\sqrt{1+|\nabla u|^2}},\nabla\varphi\right\rangle\,dx. \end{align*} Integrating by parts, with no boundary term because $\varphi$ is compactly supported in $U'$, gives \begin{align*} \int_{U'}\left\langle \frac{\nabla u}{\sqrt{1+|\nabla u|^2}},\nabla\varphi\right\rangle\,dx=-\int_{U'}\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)\varphi\,dx. \end{align*} Therefore the graph is stationary for area under compactly supported vertical variations exactly when \begin{align*} \operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=0. \end{align*} To see the harmonic equation inside this nonlinear equation, linearise at the flat graph $u=0$ in the direction $v$. For $u_t=tv$, \begin{align*} \frac{\nabla u_t}{\sqrt{1+|\nabla u_t|^2}}=\frac{t\nabla v}{\sqrt{1+t^2|\nabla v|^2}}. \end{align*} Differentiating this vector field at $t=0$ gives \begin{align*} \frac{d}{dt}\Big|_{t=0}\frac{t\nabla v}{\sqrt{1+t^2|\nabla v|^2}}=\nabla v. \end{align*} Applying divergence then gives the linearised operator \begin{align*} \operatorname{div}(\nabla v)=\Delta v. \end{align*} Thus harmonic functions are the infinitesimal model for minimal graphs: near a flat graph, the first-order part of the minimal surface equation is $\Delta v=0$. [/example] ## Stability, Compactness, And Regularity Once stationarity is understood, which stationary objects are visible in variational problems? A stationary point may be unstable, and a sequence of stationary objects may converge only after allowing weak notions and singular sets. The course therefore moves from first variation to second variation, compactness, and regularity. [definition: Stability] Let $X\in\mathcal A$ be stationary for a twice differentiable functional $\mathcal F:\mathcal A\to\mathbb R$, and let $\mathcal V_X$ be the [vector space](/page/Vector%20Space) of compactly supported admissible variation fields at $X$. The second variation at $X$ is the quadratic form $\delta^2\mathcal F_X:\mathcal V_X\to\mathbb R$ defined by \begin{align*} \delta^2\mathcal F_X(V)=\frac{d^2}{dt^2}\Big|_{t=0}\mathcal F[X_t], \end{align*} where $(X_t)$ is any admissible variation with initial variation field $V$. The stationary object $X$ is stable if $\delta^2\mathcal F_X(V)\ge 0$ for every $V\in\mathcal V_X$. [/definition] Stability is the analytic remnant of being a local minimiser, but the definition is still phrased in terms of all compactly supported variations. To use it for estimates, one must convert nonnegativity of the second variation into an inequality involving a [test function](/page/Test%20Function) and the curvature of the hypersurface. For a two-sided minimal hypersurface, normal variations have the form $\phi\nu$, so the stability condition becomes a scalar coercive inequality. [quotetheorem:5659] [citeproof:5659] This inequality is the prototype for many estimates in the course: geometric curvature terms are controlled by an analytic energy term. Each hypothesis has a specific role. Two-sidedness gives a globally defined unit normal $\nu$, so normal variations can be written as $\phi\nu$; for a one-sided hypersurface the corresponding statement must be formulated on the double cover or using sections of the normal bundle. Minimality removes the first variation term, while compact support eliminates boundary contributions that would otherwise require separate boundary conditions. Stability is also essential. An unstable minimal surface can satisfy $H=0$ while admitting a compactly supported variation with negative second variation, so the displayed inequality fails for a suitable test function. The theorem therefore gives an estimate for stable stationary hypersurfaces, not for arbitrary minimal hypersurfaces, and it is a scalar hypersurface statement rather than the full higher-codimension [second variation formula](/theorems/2729). Later, Bochner identities play the same role for harmonic maps by converting curvature assumptions into differential inequalities for $|du|^2$. [example: Why Stability Matters] For the catenoid $\Sigma\subset\mathbb R^3$, minimality gives $H=0$, but index one means that the second variation quadratic form has a negative direction. Thus there is a compactly supported smooth function $\phi$ on $\Sigma$ such that the normal variation field $V=\phi\nu$ satisfies \begin{align*} \delta^2\operatorname{Area}_\Sigma(\phi\nu)=\int_\Sigma \left(|\nabla_\Sigma \phi|^2-|A|^2\phi^2\right)\,d\operatorname{vol}_\Sigma<0. \end{align*} Here the ambient Ricci term is absent because $\operatorname{Ric}_{\mathbb R^3}(\nu,\nu)=0$. The inequality says exactly that, for this choice of $\phi$, the curvature contribution $|A|^2\phi^2$ near the neck is large enough to dominate the gradient energy $|\nabla_\Sigma\phi|^2$. Hence the catenoid is stationary but not stable. For an affine plane $P\subset\mathbb R^3$, the second fundamental form vanishes, so $A=0$. The ambient Ricci curvature also vanishes, so for every $\phi\in C_c^\infty(P)$ one has \begin{align*} \int_P \left(|A|^2+\operatorname{Ric}_{\mathbb R^3}(\nu,\nu)\right)\phi^2\,d\operatorname{vol}_P=\int_P (0+0)\phi^2\,d\operatorname{vol}_P. \end{align*} The right-hand side is \begin{align*} \int_P (0+0)\phi^2\,d\operatorname{vol}_P=0. \end{align*} Since \begin{align*} 0\le \int_P |\nabla \phi|^2\,d\operatorname{vol}_P, \end{align*} the stability inequality holds for planes. Thus $H=0$ is only the stationarity equation; stability is the extra condition that rules out negative second variation directions like the catenoid neck variation. [/example] ## Weak Objects And Singular Sets Why does a course on smooth variational equations need weak spaces and measure-theoretic language? Minimising sequences may lose smooth convergence, concentrate energy, or develop singularities. The natural compactness theorems require Sobolev maps, varifolds, currents, and measure estimates, even when the final [regularity theorem](/theorems/2750) recovers smoothness away from a small exceptional set. [definition: Weak Harmonic Map] Let $(M,g)$ and $(N,h)$ be smooth Riemannian manifolds, and suppose $N$ is isometrically embedded in $\mathbb R^q$. A map $u\in W^{1,2}(M;\mathbb R^q)$ with $u(x)\in N$ for a.e. $x\in M$ is weakly harmonic if it is stationary for the Dirichlet energy under compactly supported variations through maps into $N$. [/definition] This definition lets the course apply direct methods from the calculus of variations, but it also exposes the central obstruction: [weak convergence](/page/Weak%20Convergence) preserves energy bounds more readily than it preserves smoothness or pointwise membership in a nonlinear equation. A sequence of maps can concentrate energy at isolated points, and a sequence of submanifolds can converge with multiplicity or develop a lower-dimensional singular set. Regularity is therefore not an assumption built into the objects; it is obtained by combining compactness with estimates that rule out concentration at sufficiently small scales. [remark: Regularity Philosophy For The Course] Minimal hypersurfaces and harmonic maps satisfy elliptic systems whose smooth stationary solutions obey local estimates controlled by energy, curvature, and stability. Under the additional hypotheses appropriate to each theory, weak limits are smooth on an open regular set, while singular sets are constrained by dimension, monotonicity, or energy concentration. [/remark] Monotonicity formulas identify scale-invariant quantities, compactness theorems extract weak or varifold limits, epsilon-regularity upgrades small energy to smooth control, and blow-up arguments classify tangent objects and restrict singular behaviour. [remark: Elliptic Emphasis] Ricci flow and mean curvature flow are parabolic theories in which the geometric object evolves in time. Here the main technical emphasis is instead on elliptic variational equations: first variation gives a stationary equation, second variation gives stability information, and compactness theory describes limits of stationary or minimising objects. [/remark] ## Roadmap Of The Lectures How will the two halves of the course fit together? The lectures first build the theory of minimal surfaces from the area functional, then use the same variational pattern to study harmonic maps. The point is not to present two unrelated subjects, but to develop a shared toolkit for geometric Euler--Lagrange equations. The minimal surface part begins with smooth hypersurfaces, normal variations, the mean curvature vector, and the first variation formula. It then derives the minimal graph equation, studies second variation and the Jacobi operator, and introduces stability, Morse index, and nullity. From there it turns to Bernstein-type rigidity, curvature estimates, varifold compactness, Allard regularity, Plateau-type existence, and the structure of singular sets. The harmonic map part begins with the Dirichlet energy, the tension field, and the coordinate form of the harmonic map equation. It then develops Bochner identities, consequences of nonpositive target curvature, weak compactness in $W^{1,2}$, partial regularity, bubbling phenomena, and the Eells--Sampson existence theorem. By the end of the course, minimal surfaces and harmonic maps should read as two expressions of the same variational principle: geometry determines an energy, stationarity gives an elliptic equation, and compactness plus regularity explain what kinds of limits the equation permits. The introduction has framed the central variational theme of the course: geometry becomes an energy, stationarity becomes an elliptic equation, and compactness and regularity determine what can arise in the limit. The next chapter begins making that theme concrete by turning the search for least-area surfaces into the first variational problem of the course. # 1. Variational Geometry Of Area This chapter begins the course by turning the geometric problem of finding surfaces of least area into a variational problem. The basic objects are smooth immersed submanifolds, their area measured by [Hausdorff measure](/page/Hausdorff%20Measure), and the response of area under compactly supported deformations. The main outcome is the first variation formula: the mean curvature vector is the negative gradient of the area functional. Minimal graphs then translate the same geometric condition into a quasilinear elliptic PDE in divergence form. ## Smooth Hypersurfaces And The Area Functional The first question is how to measure the area of a curved $m$-dimensional object sitting inside Euclidean space or, more generally, inside a Riemannian manifold. The correct measure must agree with ordinary $m$-dimensional volume in parametrisations, but must also be intrinsic enough to survive changes of coordinates. We first name the class of parametrised geometric objects whose tangent spaces carry this induced volume. [definition: Smooth Immersed Submanifold] Let $(N^{n},g)$ be a smooth Riemannian manifold. A smooth immersed $m$-dimensional submanifold is a smooth manifold $M^m$ together with a smooth immersion $F:M\to N$. [/definition] The immersion condition means that $dF_p:T_pM\to T_{F(p)}N$ is injective for every $p\in M$. Pulling back the ambient metric gives a Riemannian metric $F^*g$ on $M$, so the next step is to use that metric to define the quantity whose critical points we will study. [definition: Area Functional] Let $M^m$ be a smooth manifold, let $(N^n,g)$ be a smooth Riemannian manifold, and let $K\subset M$ be compact. The area functional over $K$ is the map \begin{align*} \mathcal A_K:\{F:M\to N\mid F \text{ is a smooth immersion}\}\to [0,\infty) \end{align*} defined by \begin{align*} \mathcal A_K[F] := \int_K d\mu_{F^*g}. \end{align*} [/definition] When $F$ is an embedding and $\Sigma=F(M)$, this agrees locally with $\mathcal H^m(\Sigma)$ computed using the Riemannian distance induced by $g$. This definition is local on the parameter domain, while the Hausdorff measure description is local on the image. The equality between the two viewpoints is the [area formula](/theorems/3075), and a flat example checks that the normalisation matches Euclidean geometry. [example: Plane Has Euclidean Area] Let $F:U\subset \mathbb R^m\to \mathbb R^{m+1}$ be the flat parametrisation $F(x)=(x,0)$, where $\mathbb R^{m+1}$ has the Euclidean metric. For the standard coordinate vector $\partial_i$ on $U$, \begin{align*} dF_x(\partial_i)=e_i \end{align*} for each $1\leq i\leq m$, where $e_1,\ldots,e_m$ are the first $m$ standard basis vectors in $\mathbb R^{m+1}$. Therefore the induced metric has entries \begin{align*} (F^*g)_{ij}=g_{\mathbb R^{m+1}}(dF_x(\partial_i),dF_x(\partial_j)). \end{align*} Substituting $dF_x(\partial_i)=e_i$ and $dF_x(\partial_j)=e_j$ gives \begin{align*} (F^*g)_{ij}=g_{\mathbb R^{m+1}}(e_i,e_j)=\delta_{ij}. \end{align*} Thus the metric matrix is $I_m$. Since $\det(I_m)=1$, the induced volume density is \begin{align*} d\mu_{F^*g}=\sqrt{\det(I_m)}\,d\mathcal L^m=\sqrt{1}\,d\mathcal L^m=d\mathcal L^m. \end{align*} Hence, for any compact measurable $K\subset U$, \begin{align*} \mathcal A_K[F]=\int_K d\mu_{F^*g}=\int_K d\mathcal L^m=\mathcal L^m(K). \end{align*} In particular, when the chosen domain of integration is $U$ itself and the integral is finite, $\mathcal A_U[F]=\mathcal L^m(U)$. A flat parametrised sheet is therefore measured by ordinary [Lebesgue measure](/page/Lebesgue%20Measure) on its parameter domain, fixing the normalisation of the area functional. [/example] To vary area, we need a language for deforming the immersion. The derivative of a deformation is the infinitesimal direction in which area will be differentiated. [definition: Variation Vector Field] Let $F:M\to N$ be a smooth immersion. A smooth variation of $F$ is a smooth map $F_t:M\to N$, defined for $t$ in an interval around $0$, with $F_0=F$. Its variation vector field is \begin{align*} X:M\to F^*TN,\qquad X(p) := \frac{\partial F_t(p)}{\partial t}\Big|_{t=0} \in T_{F(p)}N. \end{align*} The variation is compactly supported in $K\subset M$ if $X$ vanishes on $M\setminus K$. [/definition] The vector field $X$ is a smooth section of the pullback tangent bundle along $F$, written $X\in\Gamma(F^*TN)$. This language separates the infinitesimal deformation from any particular choice of parametrisation of the nearby maps. The geometric part of a deformation is detected by its normal component; tangential components mainly reparametrise the same image. For hypersurfaces, this reduction is only globally scalar when a consistent unit normal has been chosen. Two-sidedness supplies such a normal, and then every compactly supported normal deformation can be encoded by a single test function multiplying that normal field. This scalar encoding is needed before the first variation can be tested by arbitrary functions on the hypersurface. The following definition fixes the class of variations whose infinitesimal field is exactly a compactly supported normal speed, so later stability and criticality statements can be expressed in terms of one function $\varphi$ rather than an arbitrary ambient vector field. [definition: Normal Variation Of A Two-Sided Hypersurface] Let $F:M^m\to (N^{m+1},g)$ be a two-sided immersed hypersurface with global unit normal $\nu\in\Gamma(NM)$. A normal variation of $F$ is a smooth variation $F_t:M\to N$, defined for $t$ in an interval around $0$ with $F_0=F$, whose variation vector field is a section $X\in\Gamma(F^*TN)$ of the form \begin{align*} X = \varphi \nu \end{align*} for some smooth compactly supported function $\varphi\in C_c^\infty(M)$. [/definition] Normal variations encode genuine motion of the hypersurface through the ambient manifold. Tangential variations still matter in the derivation of the first variation formula, but for closed or compactly supported variations their contribution becomes a boundary term. ## Mean Curvature As The Area Gradient The next question is what infinitesimal quantity measures the failure of a submanifold to be critical for area. The answer is mean curvature, obtained by taking the trace of the second fundamental form. We begin with the tensor that records how tangent directions bend into the normal bundle. [definition: Second Fundamental Form] Let $F:M^m\to (N^n,g)$ be a smooth immersion, and let $NM\to M$ denote the normal bundle of the immersion. The second fundamental form is the symmetric tensor \begin{align*} A\in \Gamma(T^*M\otimes T^*M\otimes NM) \end{align*} defined by \begin{align*} A(Y,Z) := (\nabla^N_{dF(Y)}dF(Z))^\perp \end{align*} for smooth vector fields $Y,Z\in\mathfrak X(M)$. [/definition] The second fundamental form records directional normal acceleration. Since first variation of volume takes a trace over tangent directions, we now compress this tensor into the vector that will pair with the variation field. [definition: Mean Curvature Vector] Let $F:M^m\to (N^n,g)$ be a smooth immersion and let $e_1,\dots,e_m$ be a local orthonormal frame on $M$ for the induced metric. The mean curvature vector is the section $H\in\Gamma(NM)$ defined by \begin{align*} H := \sum_{i=1}^m A(e_i,e_i). \end{align*} [/definition] Some authors divide this trace by $m$. In these notes $H$ is the full trace, because that convention makes the first variation formula appear without an additional factor. The simplest test case is a totally flat surface, where there is no normal bending to trace. [example: Mean Curvature Of A Plane] Parametrize the plane by $F:\mathbb R^2\to\mathbb R^3$, $F(x,y)=(x,y,0)$. The coordinate tangent fields are \begin{align*} \partial_xF=e_1,\qquad \partial_yF=e_2, \end{align*} so the induced metric satisfies \begin{align*} g(\partial_xF,\partial_xF)=1,\qquad g(\partial_xF,\partial_yF)=0,\qquad g(\partial_yF,\partial_yF)=1. \end{align*} Thus $\partial_x,\partial_y$ form an orthonormal frame for the induced metric. In Euclidean space the standard constant vector fields have zero covariant derivative, hence \begin{align*} \nabla^{\mathbb R^3}_{\partial_xF}\partial_xF =\nabla^{\mathbb R^3}_{e_1}e_1=0,\qquad \nabla^{\mathbb R^3}_{\partial_xF}\partial_yF =\nabla^{\mathbb R^3}_{e_1}e_2=0, \end{align*} and similarly \begin{align*} \nabla^{\mathbb R^3}_{\partial_yF}\partial_xF=0,\qquad \nabla^{\mathbb R^3}_{\partial_yF}\partial_yF=0. \end{align*} Taking normal components gives \begin{align*} A(\partial_x,\partial_x)=0,\qquad A(\partial_y,\partial_y)=0. \end{align*} Therefore the mean curvature vector is \begin{align*} H=A(\partial_x,\partial_x)+A(\partial_y,\partial_y)=0+0=0. \end{align*} For any compactly supported variation field $X$, the first variation integrand is $g(H,X)=g(0,X)=0$, so the first variation of area is zero. The plane is therefore stationary because it has no normal bending to trace. [/example] The course now proves the formula that links the geometry of $H$ to the variational derivative of area. This is the foundational computation for minimal surfaces because it identifies the first-order obstruction to stationarity. [quotetheorem:5660] [citeproof:5660] The compact support hypothesis is not a technical decoration: it is what makes the local area derivative finite even when $M$ is noncompact. If the variation meets a boundary, the divergence term produces a boundary integral depending on the tangential part of $X$ and on the conormal along the boundary, so the displayed formula is no longer the whole first variation. The formula also does not say that the surface minimizes area; it only identifies the first derivative of area at the given immersion. The sign convention is important: among normal variations, area decreases fastest in the direction of $H$. This motivates the following definition, which isolates the condition that every compactly supported variation has zero first derivative. [definition: Stationary Immersion] A smooth immersion $F:M^m\to (N^n,g)$ is stationary for area if for every compactly supported smooth variation $F_t:M\to N$ through smooth immersions with variation vector field $X$, and for every compact domain $K\subset M$ with smooth boundary such that $\operatorname{supp}X\subset K^\circ$, one has \begin{align*} \frac{d}{dt}\Big|_{t=0}\mathcal A_K[F_t]=0. \end{align*} [/definition] Stationarity is a variational condition, while $H=0$ is a geometric equation. The possible gap is that stationarity tests all compactly supported variation fields, whereas $H=0$ is a pointwise statement about the immersion. To use stationarity as a geometric definition, one must know that no cancellation in the integral can hide a nonzero mean curvature vector on a small region. Compactly supported test fields allow the variation to be localized, so the integral condition can be converted into a pointwise vanishing statement. [quotetheorem:5661] [citeproof:5661] This theorem motivates the central terminology of the first half of the course. The compact support of the test fields is what makes the implication local: on a noncompact surface, stationarity is checked by perturbing one compact region at a time, not by comparing infinite total areas. The result is also insensitive to variations that are forbidden by boundary conditions; if a boundary is fixed, only variations vanishing there are allowed, and boundary stationarity becomes a separate issue. A minimal surface is therefore not required to be globally area-minimising; at this stage it means critical for area under compactly supported variations. The catenoid and plane distinction below should be read in this first-order sense, since a critical point can still be unstable or fail to minimize among distant competitors. [definition: Minimal Immersion] A smooth immersion $F:M^m\to (N^n,g)$ is minimal if its mean curvature vector satisfies \begin{align*} H=0. \end{align*} [/definition] The flat plane is the most rigid example, but the equation $H=0$ permits cancellation between different principal curvatures. The catenoid supplies the standard non-flat model in $\mathbb R^3$. [example: Catenoid As A Minimal Surface] The catenoid in $\mathbb R^3$ is parametrised by \begin{align*} F(s,\theta)=(\cosh s\cos\theta,\cosh s\sin\theta,s), \qquad (s,\theta)\in\mathbb R\times S^1. \end{align*} Its coordinate tangent vectors are \begin{align*} F_s=(\sinh s\cos\theta,\sinh s\sin\theta,1) \end{align*} and \begin{align*} F_\theta=(-\cosh s\sin\theta,\cosh s\cos\theta,0). \end{align*} The first fundamental form satisfies \begin{align*} g_{ss}=F_s\cdot F_s=\sinh^2s\cos^2\theta+\sinh^2s\sin^2\theta+1=\sinh^2s+1=\cosh^2s. \end{align*} Also, \begin{align*} g_{s\theta}=F_s\cdot F_\theta=-\sinh s\cosh s\cos\theta\sin\theta+\sinh s\cosh s\sin\theta\cos\theta=0, \end{align*} and \begin{align*} g_{\theta\theta}=F_\theta\cdot F_\theta=\cosh^2s\sin^2\theta+\cosh^2s\cos^2\theta=\cosh^2s. \end{align*} Thus the metric matrix is diagonal, with inverse coefficients \begin{align*} g^{ss}=\cosh^{-2}s,\qquad g^{s\theta}=0,\qquad g^{\theta\theta}=\cosh^{-2}s. \end{align*} A unit normal is obtained from the cross product \begin{align*} F_s\times F_\theta=(-\cosh s\cos\theta,-\cosh s\sin\theta,\sinh s\cosh s). \end{align*} Its length is \begin{align*} |F_s\times F_\theta|=\sqrt{\cosh^2s\cos^2\theta+\cosh^2s\sin^2\theta+\sinh^2s\cosh^2s}=\sqrt{\cosh^2s(1+\sinh^2s)}=\cosh^2s. \end{align*} Hence we may choose \begin{align*} \nu=\frac{F_s\times F_\theta}{|F_s\times F_\theta|}=\left(-\frac{\cos\theta}{\cosh s},-\frac{\sin\theta}{\cosh s},\tanh s\right). \end{align*} The second derivatives are \begin{align*} F_{ss}=(\cosh s\cos\theta,\cosh s\sin\theta,0), \end{align*} \begin{align*} F_{s\theta}=(-\sinh s\sin\theta,\sinh s\cos\theta,0), \end{align*} and \begin{align*} F_{\theta\theta}=(-\cosh s\cos\theta,-\cosh s\sin\theta,0). \end{align*} Taking dot products with $\nu$ gives the scalar second fundamental form coefficients for this choice of normal: \begin{align*} h_{ss}=F_{ss}\cdot\nu=(\cosh s\cos\theta)\left(-\frac{\cos\theta}{\cosh s}\right)+(\cosh s\sin\theta)\left(-\frac{\sin\theta}{\cosh s}\right)=-(\cos^2\theta+\sin^2\theta)=-1. \end{align*} Similarly, \begin{align*} h_{s\theta}=F_{s\theta}\cdot\nu=(-\sinh s\sin\theta)\left(-\frac{\cos\theta}{\cosh s}\right)+(\sinh s\cos\theta)\left(-\frac{\sin\theta}{\cosh s}\right)=0, \end{align*} and \begin{align*} h_{\theta\theta}=F_{\theta\theta}\cdot\nu=(-\cosh s\cos\theta)\left(-\frac{\cos\theta}{\cosh s}\right)+(-\cosh s\sin\theta)\left(-\frac{\sin\theta}{\cosh s}\right)=1. \end{align*} Since the mean curvature vector is the metric trace of the second fundamental form, \begin{align*} H=(g^{ss}h_{ss}+2g^{s\theta}h_{s\theta}+g^{\theta\theta}h_{\theta\theta})\nu. \end{align*} Substituting the coefficients gives \begin{align*} H=\left(\frac{-1}{\cosh^2s}+2\cdot 0\cdot 0+\frac{1}{\cosh^2s}\right)\nu=0\cdot\nu=0. \end{align*} Equivalently, the principal curvatures are $-\operatorname{sech}^2s$ and $\operatorname{sech}^2s$, so their trace is zero. Thus the catenoid is a non-flat minimal surface in $\mathbb R^3$: its two nonzero principal curvatures cancel in the mean curvature trace. [/example] The catenoid illustrates why the theory is richer than local flatness. Minimality is a nonlinear balance of curvatures, not the assertion that every normal curvature vanishes. ## Minimal Graphs And The Minimal Surface Equation The final question in this chapter is how the geometric equation $H=0$ becomes an elliptic PDE when the surface is written as a graph. This representation is local for any hypersurface transverse to a fixed direction, and it is the bridge from differential geometry to PDE estimates. We begin by fixing the parametrisation whose area will be varied. [definition: Graph Of A Function] Let $U\subset\mathbb R^m$ be open and let $u\in C^2(U)$. The graph of $u$ is the hypersurface parametrised by \begin{align*} F:U\to\mathbb R^{m+1},\qquad F(x)=(x,u(x)). \end{align*} [/definition] For a graph, the area functional can be written directly in terms of $u$. This turns the geometric variational problem into the calculus of variations for a scalar integral. [quotetheorem:3002] [citeproof:3002] This formula shows that minimal graphs are critical points of a convex Lagrangian depending on the gradient. The $C^1$ hypothesis is exactly what is needed for the induced metric and the classical area density to be defined pointwise. When the base domain is noncompact, the formula is used locally on compact subdomains: the integral over all of $U$ may be infinite even for smooth graphs such as an affine plane over $\mathbb R^m$, so differentiating a global area value would not define a finite variational derivative. Localising to compact subsets keeps the area finite and matches the compactly supported nature of the variations. For instance, $u(x)=|x|$ on $(-1,1)$ has a graph made from two line segments meeting at a corner; the graph has finite length, but there is no classical tangent line at $0$, and $\sqrt{1+|\nabla u|^2}$ is not a pointwise continuous density on all of $U$. Thus the theorem is not a formula for arbitrary nonsmooth graphs; such examples belong to the weak or finite-perimeter theory rather than to the smooth area functional used here. The formula is also tied to the graphical parametrisation: a round circle in $\mathbb R^2$ is not a single graph over the whole $x$-axis because most interior $x$-values have two heights, so the theorem does not measure multi-sheeted or overhanging hypersurfaces by one scalar function. Its value is that it turns the smooth graphical part of the geometric problem into a scalar variational integral, so the next task is to compute the Euler-Lagrange equation of this Lagrangian and compare it with the geometric condition $H=0$. [quotetheorem:5662] [citeproof:5662] The equation is quasilinear and elliptic as long as $|\nabla u|$ is finite. The $C^2$ assumption gives the displayed divergence a classical pointwise meaning. The function $u(x)=|x|$ on $(-1,1)$ shows what fails without enough differentiability: away from $0$ the quantity $u'/\sqrt{1+(u')^2}$ is constant on each side, but it jumps at $0$, so its derivative is not an ordinary function on all of $U$. Thus the theorem does not assert a classical PDE for corners or merely Sobolev graphs; those are handled by the weak integrated first variation. The graphical hypothesis is also essential. A vertical plane in $\mathbb R^{m+1}$ is minimal, but relative to the chosen horizontal base $\mathbb R^m$ it cannot be written as $(x,u(x))$ on any open set crossing the vertical direction; similarly, the catenoid has two heights over many points of the punctured plane. The theorem therefore gives the PDE for a single smooth sheet over a fixed domain, while general minimal immersions must be treated by the invariant condition $H=0$. Its divergence form is central because it survives in weak formulations, supports comparison principles, and gives access to regularity methods. In two variables the formula can be written in the familiar coordinates used for surfaces in $\mathbb R^3$. [example: Deriving The Minimal Graph Equation In Two Variables] Let $U\subset\mathbb R^2$, let $u\in C^2(U)$, and let $\varphi\in C_c^\infty(U)$. Choose a compact domain $K\subset U$ with smooth boundary such that $\operatorname{supp}\varphi\subset K^\circ$. For the variation $u_t=u+t\varphi$, the localized graph area is \begin{align*} \mathcal A_K[u_t]=\int_K \sqrt{1+(u_x+t\varphi_x)^2+(u_y+t\varphi_y)^2}\,d\mathcal L^2. \end{align*} By the chain rule, \begin{align*} \frac{d}{dt}\left(1+(u_x+t\varphi_x)^2+(u_y+t\varphi_y)^2\right)=2(u_x+t\varphi_x)\varphi_x+2(u_y+t\varphi_y)\varphi_y. \end{align*} Applying the derivative of $r^{1/2}$ to this positive quantity gives \begin{align*} \frac{d}{dt}\sqrt{1+(u_x+t\varphi_x)^2+(u_y+t\varphi_y)^2}=\frac{(u_x+t\varphi_x)\varphi_x+(u_y+t\varphi_y)\varphi_y}{\sqrt{1+(u_x+t\varphi_x)^2+(u_y+t\varphi_y)^2}}. \end{align*} Evaluating at $t=0$ gives the first variation \begin{align*} \frac{d}{dt}\Big|_{t=0}\mathcal A_K[u_t]=\int_K \frac{u_x\varphi_x+u_y\varphi_y}{\sqrt{1+u_x^2+u_y^2}}\,d\mathcal L^2. \end{align*} Set \begin{align*} P=\frac{u_x}{\sqrt{1+u_x^2+u_y^2}}. \end{align*} Set also \begin{align*} Q=\frac{u_y}{\sqrt{1+u_x^2+u_y^2}}. \end{align*} Then the first variation can be written as \begin{align*} \frac{d}{dt}\Big|_{t=0}\mathcal A_K[u_t]=\int_K P\varphi_x\,d\mathcal L^2+\int_K Q\varphi_y\,d\mathcal L^2. \end{align*} Since $\varphi$ vanishes near $\partial K$, [integration by parts](/theorems/2098) has no boundary term and gives \begin{align*} \int_K P\varphi_x\,d\mathcal L^2=-\int_K \varphi P_x\,d\mathcal L^2. \end{align*} The same argument in the $y$-variable gives \begin{align*} \int_K Q\varphi_y\,d\mathcal L^2=-\int_K \varphi Q_y\,d\mathcal L^2. \end{align*} Adding the two identities, \begin{align*} \frac{d}{dt}\Big|_{t=0}\mathcal A_K[u_t]=-\int_K \varphi(P_x+Q_y)\,d\mathcal L^2. \end{align*} Substituting the definitions of $P$ and $Q$ gives \begin{align*} \frac{d}{dt}\Big|_{t=0}\mathcal A_K[u_t]=-\int_K \varphi\left(\partial_x\left(\frac{u_x}{\sqrt{1+u_x^2+u_y^2}}\right)+\partial_y\left(\frac{u_y}{\sqrt{1+u_x^2+u_y^2}}\right)\right)\,d\mathcal L^2. \end{align*} Stationarity for every compactly supported $\varphi$ is therefore equivalent to \begin{align*} \partial_x\left(\frac{u_x}{\sqrt{1+u_x^2+u_y^2}}\right)+\partial_y\left(\frac{u_y}{\sqrt{1+u_x^2+u_y^2}}\right)=0 \end{align*} throughout $U$. This is the two-variable minimal surface equation in divergence form. [/example] Expanding the divergence gives the nondivergence form \begin{align*} (1+u_y^2)u_{xx}-2u_xu_yu_{xy}+(1+u_x^2)u_{yy}=0. \end{align*} The divergence form is usually preferred for weak compactness, while the nondivergence form displays ellipticity through the coefficient matrix. A final check is that the PDE recovers the flat examples from the start of the chapter. [example: Affine Functions Give Minimal Graphs] Let $u(x)=a\cdot x+b$ on $U\subset\mathbb R^m$, where $a=(a_1,\ldots,a_m)\in\mathbb R^m$ and $b\in\mathbb R$. Writing $x=(x_1,\ldots,x_m)$, we have \begin{align*} u(x)=\sum_{i=1}^m a_i x_i+b. \end{align*} For each $1\leq i\leq m$, \begin{align*} \partial_i u=\partial_i\left(\sum_{j=1}^m a_jx_j+b\right)=a_i, \end{align*} so \begin{align*} \nabla u=(a_1,\ldots,a_m)=a \end{align*} and \begin{align*} |\nabla u|^2=\sum_{i=1}^m a_i^2=|a|^2. \end{align*} Therefore \begin{align*} \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} =\frac{a}{\sqrt{1+|a|^2}} =\left(\frac{a_1}{\sqrt{1+|a|^2}},\ldots,\frac{a_m}{\sqrt{1+|a|^2}}\right). \end{align*} Each component is constant on $U$, hence \begin{align*} \operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) =\sum_{i=1}^m \partial_i\left(\frac{a_i}{\sqrt{1+|a|^2}}\right) =\sum_{i=1}^m 0 =0. \end{align*} By *[Minimal Surface Equation For Graphs](/theorems/5662)*, the graph of $u$ is minimal. Thus affine graphs are exactly tilted planes in this graphical setting, and the PDE recovers the geometric fact that planes have zero mean curvature. [/example] This completes the first variational dictionary of the course: area is measured by $\mathcal H^m$ or by the induced volume form, first variation is governed by $H$, and minimal graphs solve a nonlinear elliptic equation. The same dictionary is the smooth starting point for Plateau's problem and for geometric measure theory, where minimisers may develop singularities and must be represented by currents, varifolds, or finite-perimeter sets rather than by smooth parametrisations. It also foreshadows mean curvature flow, whose evolution equation moves a hypersurface in the negative gradient direction of area. The next chapter differentiates area a second time, introducing stability, the Jacobi operator, and the Morse-theoretic structure of minimal hypersurfaces. The first variation has identified minimal hypersurfaces as critical points of area, so the natural next question is how area responds to perturbations beyond first order. That leads to stability, the Jacobi operator, and the spectral structure governing whether a minimal hypersurface is locally minimizing or merely stationary. # 2. Second Variation, Stability, And The Jacobi Operator Continuing Chapter 1's first-variation dictionary for area, these notes develop the variational theory of minimal hypersurfaces after the first variation formula. The prerequisites are the geometry of hypersurfaces, the mean curvature equation, [integration by parts](/theorems/210) on Riemannian manifolds, and basic elliptic spectral theory. The goal of this chapter is to turn second-order area calculations into analytic tools used in Chapter 5's curvature estimates and compactness theory, as well as in deformation theory. The first variation identifies minimal hypersurfaces as critical points of area. This chapter asks what happens at the next order: does area increase, decrease, or remain flat under nearby normal deformations? The answer is encoded by an [elliptic operator](/page/Elliptic%20Operator) on the hypersurface, the Jacobi operator, whose spectrum measures stability, deformation directions, and the first appearance of singular geometric behaviour. ## The Second Variation Problem For Two-Sided Hypersurfaces A minimal hypersurface has zero first derivative of area under compactly supported variations, so the quadratic term is the first term that distinguishes local minimisers from saddle points. For hypersurfaces the cleanest formula is obtained when the normal bundle has a globally defined unit normal; this lets every normal variation be represented by a scalar function. [definition: Two-Sided Hypersurface] Let $(M^{m+1},g)$ be a Riemannian manifold and let $\Sigma^m \subset M$ be an immersed hypersurface. The hypersurface $\Sigma$ is two-sided if there exists a globally defined smooth unit normal field $\nu$ along $\Sigma$. [/definition] Two-sidedness turns normal vector fields into scalar functions by writing $V=\phi\nu$. To state the second variation in a usable form, we therefore need a definition that records the speed function of a variation and fixes the compact support convention that removes boundary terms. [definition: Normal Variation With Speed Function] Let $\Sigma^m \subset (M^{m+1},g)$ be a two-sided hypersurface with unit normal $\nu$. A normal variation with speed function $\phi \in C_c^\infty(\Sigma)$ is a smooth family of immersions $F_t:\Sigma \to M$ satisfying \begin{align*} F_0(p)&=p, & \frac{\partial F_t}{\partial t}\bigg|_{t=0}(p)&=\phi(p)\nu(p). \end{align*} [/definition] The speed function is now the single unknown in the variation problem. The compact support condition localises the calculation and removes boundary terms; if $\Sigma$ is closed, the same formula applies with $\phi \in C^\infty(\Sigma)$. The next problem is to compute the second derivative of area using only this scalar input, the intrinsic gradient on $\Sigma$, and the curvature of the immersion. This motivates the following theorem, which is the basic second-order expansion of the area functional at a minimal hypersurface. [quotetheorem:5663] [citeproof:5663] The formula separates the cost of bending the variation, measured by $|\nabla^\Sigma\phi|^2$, from the geometric potential $|A|^2+\operatorname{Ric}_M(\nu,\nu)$. Curvature of the hypersurface and positive ambient Ricci curvature both make negative second variation more likely. Each hypothesis is doing visible work. Two-sidedness is what lets the normal variation be encoded by a single scalar function; on a one-sided hypersurface the same calculation must be written using normal bundle sections, or else passed to the two-sided cover. Minimality removes the first-order term, so for a nonminimal hypersurface the second derivative alone does not decide local minimality of area. Compact support, or closedness of $\Sigma$, is the condition that removes boundary terms; without it, the formula must be supplemented by boundary contributions determined by the chosen variation class. For later shorthand, define the quadratic functional \begin{align*} \delta^2\mathcal A:C_c^\infty(\Sigma)&\to\mathbb R, & \phi&\mapsto \frac{d^2}{dt^2}\bigg|_{t=0}\mathcal H^m(F_t(\Sigma)), \end{align*} where $F_t$ is any normal variation with speed function $\phi$. The second variation formula shows that this value depends only on $\phi$, not on the higher-order choice of variation. On an affine plane $\mathbb R^m\subset\mathbb R^{m+1}$, the induced Riemannian measure agrees with Lebesgue measure $\mathcal L^m$. [example: Plane In Euclidean Space] Let $\Sigma=\mathbb R^m\subset\mathbb R^{m+1}$ be an affine plane with a constant unit normal $\nu$. The tangent spaces of an affine plane are all the same $m$-plane, so constant tangent vector fields have ordinary Euclidean derivative equal to $0$. Hence the second fundamental form vanishes, since for constant tangent fields $X,Y$, \begin{align*} A(X,Y)=\left(\nabla^{\mathbb R^{m+1}}_X Y\right)^\perp=0. \end{align*} Euclidean space is flat, so $\operatorname{Ric}_{\mathbb R^{m+1}}(\nu,\nu)=0$. Therefore, for every $\phi\in C_c^\infty(\mathbb R^m)$, the second variation formula gives \begin{align*} \delta^2\mathcal A[\phi]=\int_{\mathbb R^m}\left(|\nabla \phi|^2-\left(|A|^2+\operatorname{Ric}_{\mathbb R^{m+1}}(\nu,\nu)\right)\phi^2\right)\,d\mathcal L^m. \end{align*} Substituting $|A|^2=0$ and $\operatorname{Ric}_{\mathbb R^{m+1}}(\nu,\nu)=0$ gives \begin{align*} \delta^2\mathcal A[\phi]=\int_{\mathbb R^m}\left(|\nabla \phi|^2-(0+0)\phi^2\right)\,d\mathcal L^m. \end{align*} Thus \begin{align*} \delta^2\mathcal A[\phi]=\int_{\mathbb R^m}|\nabla \phi|^2\,d\mathcal L^m. \end{align*} Since $|\nabla\phi|^2\ge 0$ pointwise and $d\mathcal L^m$ is a positive measure, this integral is nonnegative. Hence every compactly supported normal variation has nonnegative second variation: affine planes are stable minimal hypersurfaces in Euclidean space, and the whole quadratic form is just the Dirichlet energy of the speed function. [/example] The plane example is the model case where the potential term vanishes. The next task is to package the quadratic form into an operator, since stability and deformation theory are spectral questions. ## The Jacobi Operator And Normal Deformations The second variation form is symmetric, so it should be represented by a self-adjoint elliptic operator. This operator is the linearisation of mean curvature under normal variations, and its kernel records infinitesimal deformations through minimal hypersurfaces. [definition: Jacobi Operator] Let $\Sigma^m \subset (M^{m+1},g)$ be a two-sided minimal hypersurface with unit normal $\nu$. The Jacobi operator of $\Sigma$ is the [linear map](/page/Linear%20Map) \begin{align*} L:C^\infty(\Sigma)&\to C^\infty(\Sigma), & \phi&\mapsto \Delta_\Sigma\phi+\left(|A|^2+\operatorname{Ric}_M(\nu,\nu)\right)\phi, \end{align*} where $\Delta_\Sigma=\operatorname{div}_\Sigma\nabla^\Sigma$. [/definition] The definition packages the curvature potential from the second variation into a single elliptic operator. To use this package in stability arguments, we need the precise integration-by-parts identity relating the quadratic form to $L$. [quotetheorem:5664] [citeproof:5664] The operator form makes the analogy with Schrödinger operators explicit: $\Delta_\Sigma$ supplies the elliptic part and $|A|^2+\operatorname{Ric}_M(\nu,\nu)$ is the potential. It is also the point where boundary hypotheses matter most. Compact support, or closedness of $\Sigma$, justifies the integration by parts with no leftover term; on a compact hypersurface with boundary, the same identity depends on imposing Dirichlet, Neumann, free-boundary, or another geometric boundary condition. For instance, if $\Sigma=[0,1]\subset\mathbb R$ and $\phi$ is not required to vanish at the endpoints, then \begin{align*} \int_0^1 |\phi'|^2\,dx=-\int_0^1 \phi\phi''\,dx+\left[\phi\phi'\right]_{0}^{1}, \end{align*} so the operator expression misses the endpoint contribution unless the variation space supplies a boundary condition that kills it. Thus the operator formula is not a boundary-free statement in disguise: it is a statement about the quadratic form on the chosen variation space. The next question is what it means for a deformation to preserve minimality to first order. The operator $L$ was built from the linearisation of mean curvature, so a first-order deformation through minimal hypersurfaces should have zero linearised mean curvature. This turns the geometric problem of infinitesimal minimal deformations into the analytic problem of solving a homogeneous elliptic equation, which motivates the kernel equation for $L$. [definition: Jacobi Field] Let $\Sigma^m \subset (M^{m+1},g)$ be a two-sided minimal hypersurface. A smooth function $\phi$ on $\Sigma$ is a Jacobi field if \begin{align*} L\phi=0. \end{align*} [/definition] Jacobi fields are candidate infinitesimal ways of moving a minimal hypersurface while preserving minimality to first order. The geometric test for this interpretation is to begin with an actual family of minimal hypersurfaces and ask what equation its normal speed satisfies. Since every member of the family has zero mean curvature, differentiating the mean curvature equation should produce a linear elliptic equation. This motivates the Jacobi field equation below. [quotetheorem:5665] [citeproof:5665] This theorem explains why the kernel of $L$ is called the Jacobi space. It is not merely an analytic kernel: it is the space of first-order minimal deformations, modulo the caveat that not every infinitesimal deformation integrates to an actual family. The hypotheses also mark the limits of this interpretation. If the family is not a family of minimal hypersurfaces, differentiating its mean curvature gives an inhomogeneous equation rather than $L\phi=0$. If the tangential part of the variation is not separated off, the calculation mixes geometric motion with reparametrisation; only the normal component changes the hypersurface as a subset of $M$ to first order. Nonintegrable Jacobi fields can occur when the linearised equation has solutions but higher-order obstruction terms prevent an actual nearby family of minimal hypersurfaces. [example: Killing Fields Produce Jacobi Fields] Let $\Psi_s$ be the local flow of the Killing field $X$. Since $X$ is Killing, each $\Psi_s$ is a local isometry, so \begin{align*} \Psi_s^*g=g. \end{align*} If $F:\Sigma\to M$ is the original immersion, then $F_s=\Psi_s\circ F$ is a smooth family of immersions with variational vector field \begin{align*} \frac{\partial F_s}{\partial s}\bigg|_{s=0}=X\circ F. \end{align*} Decompose this vector field into tangential and normal parts along $\Sigma$: \begin{align*} X=X^\top+g(X,\nu)\nu. \end{align*} Thus the normal speed function of the variation is \begin{align*} \phi=g(X,\nu). \end{align*} Because $\Psi_s$ is an isometry, it preserves the Levi-Civita connection and sends second fundamental forms to second fundamental forms. Taking traces, the mean curvature vector of $\Psi_s(\Sigma)$ is the pushforward of the mean curvature vector of $\Sigma$: \begin{align*} \vec H_{\Psi_s(\Sigma)}=d\Psi_s(\vec H_\Sigma). \end{align*} Since $\Sigma$ is minimal, $\vec H_\Sigma=0$, and therefore \begin{align*} \vec H_{\Psi_s(\Sigma)}=d\Psi_s(0)=0. \end{align*} So $F_s(\Sigma)$ is a family of minimal hypersurfaces. By the Jacobi field equation for a smooth family of minimal hypersurfaces, the normal speed satisfies \begin{align*} L\phi=0. \end{align*} In $\mathbb R^{m+1}$, a translation field has the form $X(x)=a$, giving the Jacobi field \begin{align*} \phi(x)=\langle a,\nu(x)\rangle, \end{align*} and a rotation field has the form $X(x)=Bx$ with $B^\top=-B$, giving \begin{align*} \phi(x)=\langle Bx,\nu(x)\rangle. \end{align*} Thus ambient translations and rotations produce Jacobi fields on every two-sided minimal hypersurface in Euclidean space. [/example] Symmetry-generated Jacobi fields are geometrically forced. Stability, however, asks whether there are directions in which the quadratic form is negative, and this depends on the full spectrum rather than only the kernel. ## Stability, Index, Nullity, And First Eigenvalues The second variation formula turns local minimality into an inequality. The main question is whether every compactly supported normal deformation has nonnegative second variation, and if not, how many independent ways area can decrease. [definition: Stability] Let $\Sigma^m \subset (M^{m+1},g)$ be a two-sided minimal hypersurface. The hypersurface $\Sigma$ is stable if \begin{align*} \int_\Sigma \left(|\nabla^\Sigma \phi|^2-\left(|A|^2+\operatorname{Ric}_M(\nu,\nu)\right)\phi^2\right)\,d\mathcal H^m\ge 0 \end{align*} for every $\phi\in C_c^\infty(\Sigma)$. [/definition] The definition is exactly the nonnegativity of the second variation for compactly supported normal variations. It is a local minimising condition to second order, and rewriting it gives the estimate used later in cutoff and curvature arguments: \begin{align*} \int_\Sigma \left(|A|^2+\operatorname{Ric}_M(\nu,\nu)\right)\phi^2\,d\mathcal H^m \le \int_\Sigma |\nabla^\Sigma \phi|^2\,d\mathcal H^m. \end{align*} This inequality is the main analytic output of stability. It allows curvature terms to be controlled by derivatives of test functions, which is the entry point for cutoff arguments and compactness theorems. Here stability is indispensable: without nonnegative second variation, the potential term can dominate the gradient term for some test function, and the displayed estimate is false. On the standard catenoid in $\mathbb R^3$, a compactly supported cutoff of the neck-size variation gives a test function for which $\int_\Sigma |A|^2\phi^2\,d\mathcal H^2$ exceeds $\int_\Sigma |\nabla^\Sigma\phi|^2\,d\mathcal H^2$, so the stability inequality fails. Compact support is also part of the statement, because on noncompact hypersurfaces it prevents uncontrolled behaviour at infinity and makes integration by parts legitimate. The inequality is not a pointwise curvature bound; it is an integral estimate, and its strength comes from inserting carefully chosen cutoff functions. [example: Instability Of The Catenoid] Let $\Sigma\subset\mathbb R^3$ be the standard catenoid, parametrised by $F(t,\theta)=(\cosh t\cos\theta,\cosh t\sin\theta,t)$ for $t\in\mathbb R$ and $\theta\in[0,2\pi)$. Since $\operatorname{Ric}_{\mathbb R^3}=0$, the Jacobi operator is $L=\Delta_\Sigma+|A|^2$. The induced metric is \begin{align*} g_\Sigma=\cosh^2 t\,(dt^2+d\theta^2). \end{align*} Thus, for a rotationally symmetric function $u(t)$, \begin{align*} \Delta_\Sigma u=\frac{1}{\cosh^2 t}u''(t). \end{align*} For the standard catenoid, \begin{align*} |A|^2=2\cosh^{-4}t. \end{align*} Therefore $Lu=0$ is equivalent to \begin{align*} 0=\frac{1}{\cosh^2 t}u''(t)+\frac{2}{\cosh^4 t}u(t). \end{align*} Multiplying by $\cosh^2 t$ gives \begin{align*} 0=u''(t)+2\operatorname{sech}^2t\,u(t). \end{align*} Changing the neck size gives the rotational Jacobi field \begin{align*} u(t)=1-t\tanh t. \end{align*} Indeed, \begin{align*} u'(t)=-\tanh t-t\operatorname{sech}^2t. \end{align*} Differentiating once more, \begin{align*} u''(t)=-\operatorname{sech}^2t-\operatorname{sech}^2t+2t\operatorname{sech}^2t\tanh t. \end{align*} Hence \begin{align*} u''(t)=-2\operatorname{sech}^2t+2t\operatorname{sech}^2t\tanh t. \end{align*} Substituting $u(t)=1-t\tanh t$ into the Jacobi equation gives \begin{align*} u''(t)+2\operatorname{sech}^2t\,u(t)=\left(-2\operatorname{sech}^2t+2t\operatorname{sech}^2t\tanh t\right)+2\operatorname{sech}^2t(1-t\tanh t). \end{align*} Expanding the last expression, \begin{align*} u''(t)+2\operatorname{sech}^2t\,u(t)=-2\operatorname{sech}^2t+2t\operatorname{sech}^2t\tanh t+2\operatorname{sech}^2t-2t\operatorname{sech}^2t\tanh t. \end{align*} The two constant terms cancel and the two $t\tanh t$ terms cancel, so \begin{align*} u''(t)+2\operatorname{sech}^2t\,u(t)=0. \end{align*} Also $u(0)=1$, while $u(t)\to-\infty$ as $|t|\to\infty$, so $u$ changes sign. Choose $T>0$ with $u(T)=0$. Since $u$ is even, also $u(-T)=0$. On the compact annulus $\{|t|<T\}$, the function $u$ is a nonzero Dirichlet solution of $Lu=0$, so the first Dirichlet eigenvalue of $L$ on this annulus is $0$. If the annulus is enlarged slightly, domain monotonicity for Dirichlet eigenvalues makes the top Dirichlet eigenvalue of $L$ positive. For a corresponding eigenfunction $v$ with $Lv=\lambda v$ and $\lambda>0$, the operator form of the second variation gives \begin{align*} \delta^2\mathcal A[v]=-\int_\Sigma vLv\,d\mathcal H^2. \end{align*} Using $Lv=\lambda v$, \begin{align*} \delta^2\mathcal A[v]=-\lambda\int_\Sigma v^2\,d\mathcal H^2. \end{align*} Because $\lambda>0$ and $v$ is not identically zero, this is strictly negative. Extending the Dirichlet eigenfunction by zero gives a compactly supported test function on the complete catenoid with negative second variation. Thus the standard catenoid is not stable. The *classical rotational spectral analysis of the catenoid Jacobi operator* shows that this negative direction is the only independent one, so the catenoid has Morse index one. [/example] The catenoid shows that minimality is not the same as stability. To measure instability systematically, we count the maximal [dimension of subspaces](/theorems/375) on which the second variation is negative. [definition: Morse Index] Let $\Sigma^m \subset (M^{m+1},g)$ be a two-sided minimal hypersurface. The Morse index of $\Sigma$ is the maximal dimension of a vector subspace $E\subset C_c^\infty(\Sigma)$ such that \begin{align*} \delta^2\mathcal A[\phi]<0 \end{align*} for every nonzero $\phi\in E$. [/definition] A stable hypersurface has Morse index zero, while finite index means that instability is confined to finitely many independent directions. Once negative directions have been counted, the remaining degenerate directions need separate language because they preserve area to second order without being negative. [definition: Nullity] Let $\Sigma^m \subset (M^{m+1},g)$ be a closed two-sided minimal hypersurface. The nullity of $\Sigma$ is \begin{align*} \dim\ker L. \end{align*} [/definition] Nullity counts infinitesimal minimal deformations on a closed hypersurface. To connect index and nullity to computable quantities, we now use the spectral theorem for the self-adjoint elliptic operator $L$. [quotetheorem:5666] [citeproof:5666] This theorem explains the phrase "positive first eigenvalue" in geometric terms: for this sign convention, a positive top eigenvalue produces an area-decreasing normal direction. Stability is therefore equivalent to the absence of a positive first eigenvalue. Closedness is what allows this clean discrete list of eigenvalues: compact elliptic theory gives a real spectrum with finite-dimensional eigenspaces and no essential spectrum. On a noncompact hypersurface, $L$ may have continuous or essential spectrum, and stability is usually formulated through the quadratic form on $C_c^\infty(\Sigma)$ rather than by counting eigenvalues alone. In that setting finite index still counts negative directions of the quadratic form, but it need not be recoverable from a simple ordered sequence $\lambda_1\ge\lambda_2\ge\cdots$. [example: Equatorial Sphere In A Round Sphere] Let $\Sigma=S^m\subset S^{m+1}$ be the equatorial sphere in the unit round sphere. The equator is totally geodesic, so $A=0$. The unit round sphere satisfies $\operatorname{Ric}_{S^{m+1}}=m g$, and since $\nu$ is a unit normal, \begin{align*} \operatorname{Ric}_{S^{m+1}}(\nu,\nu)=m g(\nu,\nu)=m. \end{align*} Therefore, for every $\phi\in C^\infty(S^m)$, \begin{align*} L\phi=\Delta_{S^m}\phi+\left(|A|^2+\operatorname{Ric}_{S^{m+1}}(\nu,\nu)\right)\phi. \end{align*} Substituting $|A|^2=0$ and $\operatorname{Ric}_{S^{m+1}}(\nu,\nu)=m$ gives \begin{align*} L\phi=\Delta_{S^m}\phi+m\phi. \end{align*} For a constant function $\phi\equiv c$, the intrinsic gradient vanishes: \begin{align*} \nabla^{S^m}\phi=0. \end{align*} Hence, using $\Delta_{S^m}=\operatorname{div}_{S^m}\nabla^{S^m}$, \begin{align*} \Delta_{S^m}\phi=\operatorname{div}_{S^m}(0)=0. \end{align*} Thus \begin{align*} L\phi=0+mc=mc. \end{align*} So constants are eigenfunctions of $L$ with eigenvalue $m>0$. The operator form of the second variation gives \begin{align*} \delta^2\mathcal A[\phi]=-\int_{S^m}\phi L\phi\,d\mathcal H^m. \end{align*} Substituting $\phi=c$ and $L\phi=mc$ gives \begin{align*} \delta^2\mathcal A[\phi]=-\int_{S^m}c(mc)\,d\mathcal H^m. \end{align*} Since $c(mc)=mc^2$ is constant, \begin{align*} \delta^2\mathcal A[\phi]=-mc^2\mathcal H^m(S^m). \end{align*} If $c\ne 0$, then $m>0$, $c^2>0$, and $\mathcal H^m(S^m)>0$, so \begin{align*} \delta^2\mathcal A[\phi]<0. \end{align*} This negative direction is the latitude direction. Write the equator as $\{x_{m+2}=0\}\subset S^{m+1}\subset\mathbb R^{m+2}$, with unit normal $\nu=e_{m+2}$. The variation \begin{align*} F_t(p)=\cos t\,p+\sin t\,e_{m+2} \end{align*} has derivative \begin{align*} \frac{\partial F_t}{\partial t}=-\sin t\,p+\cos t\,e_{m+2}. \end{align*} At $t=0$ this becomes \begin{align*} \frac{\partial F_t}{\partial t}\bigg|_{t=0}=e_{m+2}=\nu. \end{align*} Thus its normal speed function is $\phi\equiv 1$. The induced metric on the latitude sphere is $(\cos^2 t)g_{S^m}$, so its area is \begin{align*} \mathcal H^m(F_t(S^m))=(\cos t)^m\mathcal H^m(S^m). \end{align*} Differentiating the scalar factor, \begin{align*} \frac{d}{dt}(\cos t)^m=-m(\cos t)^{m-1}\sin t. \end{align*} Differentiating again, \begin{align*} \frac{d^2}{dt^2}(\cos t)^m=m(m-1)(\cos t)^{m-2}\sin^2 t-m(\cos t)^m. \end{align*} At $t=0$, this gives \begin{align*} \frac{d^2}{dt^2}\bigg|_{t=0}(\cos t)^m=m(m-1)\cdot 1^{m-2}\cdot 0^2-m\cdot 1^m. \end{align*} Therefore \begin{align*} \frac{d^2}{dt^2}\bigg|_{t=0}(\cos t)^m=-m. \end{align*} So the latitude variation decreases area to second order. Finally, the *spherical-harmonic spectrum of the round sphere* says that the eigenvalues of $\Delta_{S^m}$ are \begin{align*} -\ell(\ell+m-1) \end{align*} for $\ell=0,1,2,\dots$. Therefore the corresponding eigenvalues of $L=\Delta_{S^m}+m$ are \begin{align*} m-\ell(\ell+m-1). \end{align*} For $\ell=0$, this eigenvalue is $m>0$, and the eigenspace consists of the constants, so its multiplicity is $1$. For $\ell=1$, the eigenvalue is \begin{align*} m-1(1+m-1)=m-m=0. \end{align*} For $\ell\ge 2$, \begin{align*} \ell(\ell+m-1)\ge 2(m+1). \end{align*} Hence \begin{align*} m-\ell(\ell+m-1)\le m-2(m+1)=-m-2<0. \end{align*} Thus $L$ has exactly one positive eigenvalue, counted with multiplicity. The equatorial sphere is unstable, and its Morse index is one. [/example] The equatorial sphere also has Jacobi fields coming from rotations of the ambient sphere, corresponding to first spherical harmonics. These lie in the kernel of $L$, while the constant direction is genuinely unstable. [remark: Boundary Conditions] For compact hypersurfaces with boundary, the same quadratic form is paired with a boundary condition. Compactly supported variations avoid boundary terms, while fixed-boundary variations impose Dirichlet conditions on $\phi$. Free-boundary minimal hypersurfaces require an additional boundary contribution, which is a separate topic because the boundary geometry contributes to stability. [/remark] The same dictionary also connects the course to several neighbouring subjects. In PDE, the Jacobi operator is a geometric Schrödinger operator whose quadratic form is analysed by the same methods used for eigenvalue problems and coercivity estimates. In Riemannian geometry, Jacobi fields parallel the role of geodesic Jacobi fields: both describe infinitesimal families of critical objects and both detect conjugacy or degeneracy. In mathematical physics, stability of minimal hypersurfaces is the static analogue of asking whether a linearised field equation has negative energy modes. The chapter's main conclusion is that the second variation is not just a formula but a dictionary. Stability is a quadratic inequality, index counts positive spectral directions of $L$, nullity is the Jacobi kernel, and geometric deformations appear as solutions of the Jacobi equation. Chapter 5's curvature estimates and compactness results use this dictionary by inserting carefully chosen cutoff functions into the stability inequality. The second variation shows how the linearized geometry controls stability, index, and nullity, and how the Jacobi equation encodes infinitesimal deformations. From there, the global rigidity of minimal graphs becomes the next test case, where these local variational ideas interact with Bernstein-type phenomena. # 3. Minimal Graphs And Bernstein Phenomena After Chapters 1 and 2 identify the minimal graph equation and the stability operator, this chapter studies the global rigidity consequences for minimal graphs. The course studies how variational principles produce nonlinear elliptic equations, how geometric quantities such as mean curvature and the second fundamental form enter those equations, and how compactness and regularity theory turn local estimates into global structure theorems. The standing prerequisites are the first and second variation formulae for area, basic elliptic estimates for divergence-form equations, Sobolev-space compactness, and the interpretation of a graph $\Gamma_u=\{(x,u(x)) : x\in U\}$ as a hypersurface in Euclidean space. This chapter turns the minimal surface equation from the preceding variational discussion into a rigidity theory for entire graphs. The central question is when a solution $u:\mathbb R^n\to\mathbb R$ of the minimal graph equation must be affine. In dimension two this is Bernstein's theorem; in higher dimensions the answer changes, and the change is governed by the regularity and stability theory of minimal hypersurfaces. The theme is that graphicality gives an elliptic equation, but the strength of that ellipticity depends on the gradient. Bounded gradient leads to a uniformly elliptic divergence-form equation and hence to interior estimates. Global rigidity then follows by combining those estimates with scaling and Liouville-type arguments. ## Bounded Gradients and Uniform Ellipticity The first problem is analytic: the minimal surface equation is elliptic, but its coefficients depend on $\nabla u$. We need to know when the equation has the uniform ellipticity needed for interior estimates. [definition: Minimal Graph Operator] Let $U\subset\mathbb R^n$ be open. The minimal graph operator is the map \begin{align*} M:C^2(U)\to C^0(U) \end{align*} defined by \begin{align*} M[u] := \operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right). \end{align*} [/definition] The equation $M[u]=0$ is the minimal surface equation for the graph $\Gamma_u := \{(x,u(x)) : x\in U\}\subset\mathbb R^{n+1}$. This operator is nonlinear because both the direction and size of $\nabla u$ affect the area element of the graph. To use elliptic PDE, we rewrite the equation in non-divergence form and inspect the coefficient matrix. [quotetheorem:5667] [citeproof:5667] The theorem isolates the main obstruction: ellipticity degenerates only when $|\nabla u|$ becomes large. Thus bounded gradient is the bridge from geometric minimality to standard interior estimates. Without a bound on $|\nabla u|$, the smallest eigenvalue \begin{align*} \frac{1}{1+|\nabla u|^2} \end{align*} may approach $0$, so the equation is no longer uniformly elliptic even though it remains elliptic at each point. In that regime the coefficient-matrix theorem gives no scale-independent Harnack inequality, Schauder estimate, or Krylov--Safonov estimate; those estimates require an ellipticity ratio bounded independently of the particular solution. The $C^2$ hypothesis is also doing real work here: the displayed non-divergence form uses pointwise second derivatives, while weaker solutions require divergence-form or varifold methods rather than this coefficient-matrix computation. [definition: Uniform Ellipticity for a Minimal Graph] Let $U\subset\mathbb R^n$ be open and let $u\in C^2(U)$ solve the minimal surface equation. The equation is uniformly elliptic on $U$ if there exists $\theta>0$ such that \begin{align*} \sum_{i,j=1}^n a_{ij}(\nabla u(x))\xi_i\xi_j\ge \theta |\xi|^2 \end{align*} for every $x\in U$ and every $\xi\in\mathbb R^n$. [/definition] The coefficient estimate above gives a quantitative version of this definition: if $|\nabla u|\le K$ on $U$, then the ellipticity constant may be taken to be $\theta=(1+K^2)^{-1}$. The same calculation also explains why a local gradient estimate is the decisive estimate in the theory. [example: Affine Entire Graphs] Let $u(x)=a\cdot x+b$ on $\mathbb R^n$, with $a=(a_1,\ldots,a_n)\in\mathbb R^n$ and $b\in\mathbb R$. Since \begin{align*} u(x)=\sum_{k=1}^n a_kx_k+b, \end{align*} we have, for each $i$, \begin{align*} \partial_{x_i}u(x)=a_i. \end{align*} Thus $\nabla u=a$ is constant. Differentiating once more gives, for every $i,j$, \begin{align*} \partial_{x_i x_j}u(x)=\partial_{x_i}(a_j)=0. \end{align*} Therefore the non-divergence form of the minimal graph equation gives \begin{align*} \sum_{i,j=1}^n a_{ij}(\nabla u)\,\partial_{x_i x_j}u=\sum_{i,j=1}^n a_{ij}(a)\,0=0, \end{align*} so $M[u]=0$. The graph is \begin{align*} \Gamma_u=\{(x,a\cdot x+b):x\in\mathbb R^n\}, \end{align*} which is the affine hyperplane in $\mathbb R^{n+1}$ with equation $x_{n+1}=a\cdot x+b$. Its coefficient matrix is constant: \begin{align*} A(a)=I-\frac{a\otimes a}{1+|a|^2}. \end{align*} If $\xi\perp a$, then $a\cdot \xi=0$, and hence \begin{align*} A(a)\xi=\xi-\frac{a(a\cdot \xi)}{1+|a|^2}=\xi. \end{align*} So every direction orthogonal to $a$ has eigenvalue $1$. If $a\ne 0$, then \begin{align*} A(a)a=a-\frac{a(a\cdot a)}{1+|a|^2}. \end{align*} Since $a\cdot a=|a|^2$, this becomes \begin{align*} A(a)a=a-\frac{|a|^2}{1+|a|^2}a. \end{align*} Combining the two scalar multiples of $a$ gives \begin{align*} A(a)a=\frac{1+|a|^2-|a|^2}{1+|a|^2}a=\frac{1}{1+|a|^2}a. \end{align*} Thus the eigenvalue in the slope direction is $(1+|a|^2)^{-1}$. Affine entire graphs solve the minimal surface equation with constant slope and zero second derivatives, so they are the flat model conclusions in Bernstein-type rigidity theorems. [/example] The affine example also shows that the size of the slope is not the issue by itself. A plane may have any constant slope; the rigidity problem asks whether the slope can vary over all of $\mathbb R^n$ while the graph remains area-stationary. ## Interior Gradient Estimates The next question is how much of the slope can be controlled from information on the height of the graph. Interior estimates answer this by bounding $|\nabla u|$ on a smaller ball in terms of the oscillation or height on a larger ball. [quotetheorem:5668] [citeproof:5668] This estimate is the analytic engine behind many Bernstein arguments. It says that if the height grows sublinearly on larger and larger balls, then the gradient becomes globally controlled, and after rescaling the surface looks flatter. The hypotheses are essential: the estimate is interior, so it gives no boundary control near $\partial B(x_0,R)$, and its right-hand side depends on the oscillation on the larger ball rather than only on $u(x_0)$. It also does not by itself prove that every entire solution is affine, since linear height growth keeps $\operatorname{osc}_{B(x_0,R)}u/R$ bounded but not small; additional Liouville or blow-down input is needed to force the slope to be constant. [example: Sublinear Entire Solutions] Let $u\in C^2(\mathbb R^n)$ solve the minimal surface equation and suppose \begin{align*} \operatorname{osc}_{B(0,R)}u=o(R) \end{align*} as $R\to\infty$. Fix $x_0\in\mathbb R^n$. For $r>2|x_0|$, the inclusion $B(x_0,r)\subset B(0,r+|x_0|)$ gives \begin{align*} \operatorname{osc}_{B(x_0,r)}u\le \operatorname{osc}_{B(0,r+|x_0|)}u. \end{align*} Dividing by $r$ and multiplying by $(r+|x_0|)/(r+|x_0|)$ gives \begin{align*} \frac{\operatorname{osc}_{B(x_0,r)}u}{r}\le \frac{\operatorname{osc}_{B(0,r+|x_0|)}u}{r+|x_0|}\cdot \frac{r+|x_0|}{r}. \end{align*} The first factor tends to $0$ by the sublinear oscillation assumption, because $r+|x_0|\to\infty$, and the second factor tends to $1$ because $x_0$ is fixed. Hence \begin{align*} \frac{\operatorname{osc}_{B(x_0,r)}u}{r}\to 0 \end{align*} as $r\to\infty$. Applying the *Moser Interior Gradient Estimate* on $B(x_0,r)$ gives \begin{align*} \sqrt{1+|\nabla u(x_0)|^2}\le C\exp\left(C\frac{\operatorname{osc}_{B(x_0,r)}u}{r}\right), \end{align*} where $C=C(n)$ and $x_0\in B(x_0,r/2)$. Letting $r\to\infty$ gives \begin{align*} \sqrt{1+|\nabla u(x_0)|^2}\le C. \end{align*} Therefore \begin{align*} |\nabla u(x_0)|^2\le C^2-1. \end{align*} Since $x_0$ was arbitrary, $\nabla u$ is globally bounded. In any setting where the corresponding Liouville theorem says that a bounded-gradient entire minimal graph has constant gradient, write $\nabla u\equiv a$. Then, for every $x\in\mathbb R^n$, the one-variable [fundamental theorem of calculus](/theorems/632) applied to $t\mapsto u(tx)$ gives \begin{align*} u(x)-u(0)=\int_0^1 \nabla u(tx)\cdot x\,dt. \end{align*} Substituting $\nabla u(tx)=a$ gives \begin{align*} u(x)-u(0)=\int_0^1 a\cdot x\,dt. \end{align*} Since $a\cdot x$ is independent of $t$, \begin{align*} \int_0^1 a\cdot x\,dt=a\cdot x. \end{align*} Thus \begin{align*} u(x)=a\cdot x+u(0), \end{align*} so the sublinear oscillation hypothesis converts the interior estimate into global slope control, and the Liouville input then upgrades bounded slope to affine rigidity. [/example] The estimate also explains why entire graphs are more rigid than general complete minimal surfaces. A complete minimal surface can wind around in space, while a global graph has a distinguished vertical direction and a single-valued height function that can be fed into elliptic estimates. [example: Helicoid Versus Graph Obstruction] The helicoid may be parametrized by \begin{align*} X(r,t)=(r\cos t,r\sin t,t), \end{align*} where $r,t\in\mathbb R$. Projection to the horizontal plane sends this point to \begin{align*} \pi(X(r,t))=(r\cos t,r\sin t). \end{align*} Fix $r\ne 0$ and let $k\in\mathbb Z$. Since sine and cosine are $2\pi$-periodic, \begin{align*} \cos(t+2\pi k)=\cos t \end{align*} and \begin{align*} \sin(t+2\pi k)=\sin t. \end{align*} Therefore \begin{align*} \pi(X(r,t+2\pi k))=(r\cos t,r\sin t). \end{align*} But the corresponding point on the helicoid is \begin{align*} X(r,t+2\pi k)=(r\cos t,r\sin t,t+2\pi k). \end{align*} Thus the same horizontal base point $(r\cos t,r\sin t)$ occurs at the distinct heights $t+2\pi k$. An entire graph over a fixed plane $P$ with unit normal $\nu$ has the form \begin{align*} \{p+u(p)\nu:p\in P\}. \end{align*} For each $p\in P$, the line $\{p+s\nu:s\in\mathbb R\}$ meets this graph exactly at $p+u(p)\nu$. The helicoid fails this single-valuedness condition for projection to the horizontal plane, because one vertical line meets it at infinitely many distinct heights. Completeness and embeddedness therefore do not replace graphicality in Bernstein's theorem; the global one-valued projection is part of the rigidity mechanism. [/example] ## Bernstein's Theorem in Dimension Two We now ask the central global question in the first nontrivial case: if $u:\mathbb R^2\to\mathbb R$ solves the minimal surface equation everywhere, must its graph be a plane? Bernstein's theorem says yes, and its proof combines the special conformal structure of two-dimensional minimal surfaces with global graph information. [quotetheorem:5669] [citeproof:5669] The statement is stronger than a local regularity theorem: it is a global rigidity theorem. The entire-graph hypothesis cannot be removed: the catenoid and helicoid are complete non-affine minimal surfaces in $\mathbb R^3$, but neither is a single-valued graph over all of $\mathbb R^2$. The dimension-two hypothesis is also essential to this proof, because it uses the conformal structure of Riemann surfaces and the meromorphic Gauss map rather than only uniform ellipticity. In higher graph dimensions the same statement eventually fails, with non-affine entire solutions appearing for graphs over $\mathbb R^n$ when $n\ge 8$. [example: Non-Affine Minimal Surfaces Do Not Contradict Bernstein] The catenoid and helicoid are minimal surfaces in $\mathbb R^3$, but Bernstein's theorem does not apply to them because neither is a single-valued graph over all of $\mathbb R^2$. For the catenoid, use the parametrization \begin{align*} X(s,t)=(\cosh s\cos t,\cosh s\sin t,s), \end{align*} with $s,t\in\mathbb R$. Its projection to the horizontal plane is \begin{align*} \pi(X(s,t))=(\cosh s\cos t,\cosh s\sin t). \end{align*} The squared radius of this projected point is \begin{align*} |\pi(X(s,t))|^2=(\cosh s\cos t)^2+(\cosh s\sin t)^2. \end{align*} Factoring out $\cosh^2 s$ gives \begin{align*} |\pi(X(s,t))|^2=\cosh^2 s(\cos^2 t+\sin^2 t). \end{align*} Since $\cos^2 t+\sin^2 t=1$, this becomes \begin{align*} |\pi(X(s,t))|^2=\cosh^2 s. \end{align*} Because $\cosh s\ge 1$ for every $s\in\mathbb R$, every projected point has radius at least $1$. Thus the projection misses every point of the disk $\{(x_1,x_2):x_1^2+x_2^2<1\}$, so the catenoid is not a graph over all of $\mathbb R^2$. For the helicoid, use \begin{align*} Y(r,t)=(r\cos t,r\sin t,t), \end{align*} with $r,t\in\mathbb R$. Projection gives \begin{align*} \pi(Y(r,t))=(r\cos t,r\sin t). \end{align*} For every integer $k$, \begin{align*} \pi(Y(r,t+2\pi k))=(r\cos(t+2\pi k),r\sin(t+2\pi k)). \end{align*} Using the $2\pi$-periodicity of sine and cosine, this is \begin{align*} \pi(Y(r,t+2\pi k))=(r\cos t,r\sin t)=\pi(Y(r,t)). \end{align*} But the corresponding point on the helicoid is \begin{align*} Y(r,t+2\pi k)=(r\cos t,r\sin t,t+2\pi k). \end{align*} If $k\ne 0$, then $t+2\pi k\ne t$, so the same horizontal point occurs at different heights. Therefore the helicoid is not single-valued over the horizontal plane. These surfaces show that local minimality and nonzero curvature are compatible; Bernstein rigidity requires the stronger condition of being an entire graph. [/example] A second route to Bernstein's theorem is more PDE-oriented: derive global gradient control and then apply a Liouville theorem to derivatives of $u$. This viewpoint is the one that generalises most naturally to the higher-dimensional estimates, even though the final rigidity threshold is not the same in every dimension. [remark: Role of the Dimension Two Hypothesis] For $n=2$, conformal coordinates and the meromorphic Gauss map give tools that are unavailable for general hypersurfaces in $\mathbb R^{n+1}$. The theorem is therefore not merely an elliptic PDE statement with $n$ as a harmless parameter. The eventual failure in high dimension reflects the existence of stable singular minimal cones. [/remark] ## Stable Cones and the Higher-Dimensional Threshold The higher-dimensional Bernstein problem asks for which $n$ every entire minimal graph $u:\mathbb R^n\to\mathbb R$ is affine. The answer is affirmative for $n\le 7$ and false for $n\ge 8$, and the dividing line is tied to whether stable minimal cones can have singularities. [definition: Minimal Cone] A minimal cone in $\mathbb R^{m}$ is a cone $C\subset\mathbb R^m$ that is stationary for the area functional with respect to $\mathcal H^{m-1}$ away from its vertex. [/definition] Minimal cones arise as blow-down limits of complete minimal hypersurfaces and as blow-up limits near possible singularities. To see why they affect Bernstein rigidity, we need a concrete non-flat cone that is minimal and later stable. [definition: Simons Cone] The Simons cone is the hypersurface \begin{align*} C = \{(x,y)\in\mathbb R^4\times\mathbb R^4 : |x|=|y|\}\subset\mathbb R^8. \end{align*} [/definition] This cone has dimension seven as a hypersurface in $\mathbb R^8$ and has an isolated singularity at the origin. The next issue is not only whether it is minimal, but whether the second variation allows it to occur as a stable limiting object. [quotetheorem:5670] [citeproof:5670] The stability of the Simons cone is the geometric signal that dimension eight in the ambient space is different. The phrase "away from the origin" is necessary because the vertex is singular, so the classical second variation formula applies only on the smooth part of the cone; variations interacting with the vertex require the broader weak theory of stationary varifolds or currents. The symmetric case $p=q=3$ is also special: the theorem does not claim that every cone $C_{p,q}$ is stable, only that this dimension and ratio give the stable model relevant to the Bernstein threshold. In lower dimensions the corresponding product cones fail the same stability test because the angular eigenvalue and the Hardy constant no longer balance; for instance the cone over $S^1\times S^1\subset S^3$ is the singular cone in $\mathbb R^4$ given by $|x|=|y|$ with $x,y\in\mathbb R^2$, and it is unstable. Stable cones with singular vertices can appear as limits only after the critical dimension is reached, and once such cones exist, the methods proving global flatness of entire graphs lose a key regularity input. [example: The Simons Cone in Euclidean Eight-Space] Write a point of $\mathbb R^8=\mathbb R^4\times\mathbb R^4$ as $(x,y)$, and set \begin{align*} C=\{(x,y): |x|=|y|\}. \end{align*} Equivalently, \begin{align*} |x|=|y| \quad\Longleftrightarrow\quad |x|^2=|y|^2 \quad\Longleftrightarrow\quad F(x,y):=|x|^2-|y|^2=0. \end{align*} Thus $F(x,y)>0$ exactly when $|x|^2>|y|^2$, which is equivalent to $|x|>|y|$, and $F(x,y)<0$ exactly when $|x|^2<|y|^2$, which is equivalent to $|x|<|y|$. Hence the complement of $C$ is the union of the two open regions $|x|>|y|$ and $|x|<|y|$. The defining equation also makes the symmetries visible. If $Q_1,Q_2\in O(4)$, then orthogonality gives \begin{align*} |Q_1x|^2=(Q_1x)\cdot(Q_1x)=x\cdot(Q_1^\topQ_1)x=x\cdot x=|x|^2. \end{align*} Similarly, \begin{align*} |Q_2y|^2=|y|^2. \end{align*} Therefore \begin{align*} |x|=|y| \quad\Longrightarrow\quad |Q_1x|=|Q_2y|, \end{align*} so $C$ is invariant under $O(4)\times O(4)$. If $\lambda\in\mathbb R$, then \begin{align*} |\lambda x|=|\lambda|\,|x| \end{align*} and \begin{align*} |\lambda y|=|\lambda|\,|y|. \end{align*} Thus $|x|=|y|$ implies $|\lambda x|=|\lambda y|$, so $C$ is invariant under dilations and is genuinely a cone. Away from the origin, $C$ is smooth because it is the zero set of $F$ with nonzero gradient. Indeed, \begin{align*} \nabla F(x,y)=(2x,-2y). \end{align*} If $(x,y)\in C$ and $(x,y)\ne(0,0)$, then at least one of $x,y$ is nonzero, so \begin{align*} (2x,-2y)\ne(0,0). \end{align*} By the regular level set theorem, $C\setminus\{0\}$ is a smooth hypersurface of $\mathbb R^8$. On the unit sphere, the link is \begin{align*} C\cap S^7=\{(x,y): |x|=|y|,\ |x|^2+|y|^2=1\}. \end{align*} Putting $r=|x|=|y|$, the sphere equation gives \begin{align*} r^2+r^2=1. \end{align*} Hence \begin{align*} 2r^2=1. \end{align*} Since $r\ge 0$, this gives \begin{align*} r=\frac{1}{\sqrt 2}. \end{align*} Thus \begin{align*} C\cap S^7=S^3\left(\frac1{\sqrt2}\right)\times S^3\left(\frac1{\sqrt2}\right). \end{align*} The cone is obtained by radially dilating this link, and at radius $0$ the whole link collapses to the single vertex $(0,0)$, where $\nabla F=0$ and the smooth level-set description breaks down. By the *Simons Cone Stability Criterion*, this smooth part is minimal; the singular vertex is the feature that makes the Simons cone the model high-dimensional obstruction to Bernstein rigidity. [/example] The examples above show why a dimension threshold is unavoidable: in low dimensions tangent-cone regularity rules out the singular models that would appear at infinity, while in higher dimensions the Simons cone supplies a stable singular obstruction. Bernstein rigidity is therefore not just a calculation with the minimal graph equation; it depends on whether blow-down limits of entire minimal graphs can contain stable singular cones. The course-level result packages this threshold for entire minimal graphs. [quotetheorem:5671] [citeproof:5671] This result closes the Bernstein discussion by identifying the obstruction with the singularity theory of stable minimal hypersurfaces. Each hypothesis has a sharp role. The restriction $n\le 7$ is necessary because the Bombieri--De Giorgi--Giusti examples give non-affine smooth entire graphs when $n\ge 8$. The word "entire" is necessary because local non-affine solutions of the minimal surface equation exist on bounded domains, for instance small graphical pieces of catenoids or helicoids after choosing a suitable projection. The graphical hypothesis is necessary because complete non-graphical minimal hypersurfaces can have nonzero curvature even in low dimension. The minimal graph equation is smooth and elliptic at every finite point of a smooth entire solution, yet its large-scale geometry can converge to a singular cone in high dimension. [remark: Ambient and Graph Dimensions] An entire graph over $\mathbb R^n$ is an $n$-dimensional hypersurface in $\mathbb R^{n+1}$. The first failure of Bernstein rigidity occurs for graphs over $\mathbb R^8$, hence hypersurfaces in $\mathbb R^9$. The Simons cone itself lies in $\mathbb R^8$ as a seven-dimensional cone, and it is the stable singular model that drives the regularity threshold used in the Bernstein theorem. [/remark] ## What the Bernstein Phenomenon Teaches The final question is conceptual: why should an equation as concrete as the minimal graph equation care about dimension seven? The answer is that global graph rigidity is not controlled only by local elliptic estimates; it also depends on the classification of possible tangent cones at infinity. [explanation: Analytic and Geometric Inputs] The analytic input is the interior gradient theory for minimal graphs. It converts height and oscillation information into slope control, and bounded slope converts the equation into a uniformly elliptic one. These estimates are local and scale well, so they are suited to blow-up and blow-down arguments. The geometric input is the stability theory from the second variation. Entire minimal graphs are stable because vertical translations and graphical variations fit naturally into the area-minimising framework on compact subdomains. Stable cones are therefore the correct limiting objects. If the only stable cones are planes, blow-down analysis enforces flatness; if a non-flat stable cone exists, the same limiting procedure allows non-affine behaviour at infinity. [/explanation] The chapter therefore links the first two parts of the course: first variation gives the minimal graph equation, second variation gives stability, and the Bernstein problem tests how far these two structures force global rigidity. The next compactness and regularity results will make this limiting viewpoint systematic. Minimal graphs make the first two chapters visible in a particularly rigid setting: stationarity gives the minimal surface equation, and stability constrains the possible global behavior. The next chapter shifts from rigidity to scale, introducing the monotonicity and blow-up tools needed to understand what minimal surfaces look like under magnification. # 4. Monotonicity, Blow-Up, And Tangent Cones The preceding chapters treated minimal submanifolds through first and second variation: Chapter 1 identified stationarity with vanishing mean curvature, while Chapter 2 encoded stability through the second variation and the Jacobi operator. This chapter introduces the local scale-invariant quantities that make compactness and singularity analysis possible. The guiding principle is that a minimal submanifold, when viewed at smaller and smaller scales around a point, begins to look like a cone, and the monotonicity formula is the mechanism that forces this limiting shape. The chapter has three linked goals. First, we prove the density ratio monotonicity formula for stationary minimal submanifolds in Euclidean balls. Second, we use it to construct blow-up limits and tangent cones. Third, we record the first consequences: local area bounds, removable singularities under density and stability hypotheses, and the concentration picture that will reappear for harmonic maps. ## Density Ratios In Euclidean Balls The first question is how to measure the amount of an $m$-dimensional minimal submanifold near a point in a way that does not change under rescaling. Ordinary area is not suitable, because replacing $x$ by $r^{-1}(x-x_0)$ multiplies $m$-dimensional area by $r^{-m}$. The natural quantity is therefore an area divided by the area scale $r^m$. Throughout this section $\rho$ denotes a radius, and $\theta_m=\mathcal H^m(B^m(0,1))$ denotes the Euclidean volume of the unit ball in $\mathbb R^m$. [definition: Density Ratio] Let $\Sigma^m \subset B(x_0,R) \subset \mathbb R^n$ be an $m$-dimensional rectifiable submanifold, or let $V$ be an integral $m$-varifold in $B(x_0,R)$. Write $\mu_\Sigma$ for the associated mass measure: $\mu_\Sigma=\mathcal H^m\big|_\Sigma$ in the multiplicity-one submanifold case, and $\mu_\Sigma=\|V\|$ in the varifold case. For $0<\rho<R$, the density ratio at $x_0$ and scale $\rho$ is \begin{align*} \Theta(\Sigma,x_0,\rho)=\frac{\mu_\Sigma(B(x_0,\rho))}{\theta_m \,\rho^m}. \end{align*} [/definition] This ratio compares $\Sigma$ with a flat multiplicity-one $m$-plane through $x_0$. The normalising factor $\theta_m\rho^m$ is exactly the $m$-dimensional measure of a radius $\rho$ ball in such a plane. [example: Density One Plane] Let $P\subset \mathbb R^n$ be an $m$-dimensional affine plane through $x_0$. Translation by $-x_0$ identifies $P$ with an $m$-dimensional linear subspace, so $P\cap B(x_0,\rho)$ is the radius-$\rho$ Euclidean ball inside that plane: \begin{align*} P\cap B(x_0,\rho)=\{x\in P: |x-x_0|<\rho\}. \end{align*} Since $\theta_m=\mathcal H^m(B^m(0,1))$ and $m$-dimensional Hausdorff measure scales by $\rho^m$ under dilation by $\rho$, we get \begin{align*} \mathcal H^m(P\cap B(x_0,\rho))=\mathcal H^m(B^m(0,\rho)). \end{align*} Also \begin{align*} \mathcal H^m(B^m(0,\rho))=\rho^m\mathcal H^m(B^m(0,1)). \end{align*} Therefore \begin{align*} \mathcal H^m(P\cap B(x_0,\rho))=\theta_m\rho^m. \end{align*} Substituting this into the definition of the density ratio gives \begin{align*} \Theta(P,x_0,\rho)=\frac{\mathcal H^m(P\cap B(x_0,\rho))}{\theta_m\rho^m}=\frac{\theta_m\rho^m}{\theta_m\rho^m}=1. \end{align*} Thus a multiplicity-one flat plane has density ratio $1$ at every scale. If the same plane is counted with integer multiplicity $q$, then the mass of $P\cap B(x_0,\rho)$ is $q\theta_m\rho^m$, and hence \begin{align*} \Theta(qP,x_0,\rho)=\frac{q\theta_m\rho^m}{\theta_m\rho^m}=q. \end{align*} So the density ratio records exactly the sheet multiplicity of a flat plane. [/example] The example explains why the density ratio is the correct comparison quantity. The next problem is to understand whether minimality forces this ratio to improve, worsen, or remain controlled as the radius changes. ## The Monotonicity Formula The central calculation uses stationarity with a radial vector field. Since the first variation vanishes for compactly supported variations, we can test the submanifold against a vector field which expands space away from $x_0$. The tangential divergence of this vector field records the area inside balls, while the radial normal component records the failure of the submanifold to be conical. [quotetheorem:5672] [citeproof:5672] The right-hand side is nonnegative, so the density ratio can only increase with scale. More importantly, equality across an annulus means the radial vector $x-x_0$ is tangent to $\Sigma$ almost everywhere in that annulus, which is the geometric signature of a cone. The hypotheses are part of the content. Without stationarity, the radial first-variation term need not vanish; for instance, a curved nonminimal graph can have density ratios which decrease after recentering because mean curvature contributes an uncontrolled error term. Local finite mass is also essential, since the ratio is not a finite scale-invariant quantity if infinitely many sheets accumulate in a compact ball. Finally, this exact formula is Euclidean: in a Riemannian manifold one obtains an almost-monotonicity formula with curvature-dependent error terms in sufficiently small geodesic balls, not the identity above. [remark: Role Of Stationarity] Minimality is used only through stationarity in the monotonicity formula. This is why the statement naturally belongs to the language of varifolds: singular or weak limits of smooth minimal submanifolds still satisfy the same first variation identity. Smoothness is useful for deriving the formula, but the identity survives under varifold convergence. [/remark] A useful consequence is that the limiting density exists at every point. This is the first place where monotonicity turns local geometry into a numerical invariant. [definition: Density At A Point] Let $\Sigma$ be as in the monotonicity formula. The density of $\Sigma$ at $x_0$ is \begin{align*} \Theta(\Sigma,x_0)=\lim_{\rho\downarrow 0}\Theta(\Sigma,x_0,\rho). \end{align*} [/definition] The limit exists because the density ratio is monotone and locally bounded below. At a smooth embedded point the tangent plane calculation gives density $1$, while higher density records multiplicity or singular concentration. [example: Cone Over A Geodesic Submanifold Of A Sphere] Let $N^{m-1}\subset S^{n-1}$ be a smooth minimal submanifold of the unit sphere, and set \begin{align*} C(N)=\{r y: r\ge 0,\ y\in N\}\subset \mathbb R^n. \end{align*} For $r>0$, the tangent space of $C(N)$ at $r y$ is spanned by the radial vector $y$ and the tangent space $T_yN$. If $e_1,\ldots,e_{m-1}$ is an [orthonormal basis](/page/Orthonormal%20Basis) for $T_yN$, then $y,e_1,\ldots,e_{m-1}$ is an orthonormal basis for $T_{r y}C(N)$. The second fundamental contribution in the radial direction is zero, and the link directions satisfy \begin{align*} \left(\nabla^{\mathbb R^n}_{e_i}e_i\right)^\perp_{C(N)}=\frac{1}{r}\left(\nabla^{S^{n-1}}_{e_i}e_i\right)^\perp_N. \end{align*} Thus the mean curvature vector of the cone is \begin{align*} H_{C(N)}(r y)=\frac{1}{m r}\sum_{i=1}^{m-1}\left(\nabla^{S^{n-1}}_{e_i}e_i\right)^\perp_N. \end{align*} Since \begin{align*} H_N(y)=\frac{1}{m-1}\sum_{i=1}^{m-1}\left(\nabla^{S^{n-1}}_{e_i}e_i\right)^\perp_N, \end{align*} we get \begin{align*} H_{C(N)}(r y)=\frac{m-1}{m r}H_N(y). \end{align*} Because $N$ is minimal in $S^{n-1}$, $H_N=0$, so $C(N)\setminus\{0\}$ is minimal in $\mathbb R^n\setminus\{0\}$. Now compute the density ratio at the origin. The parametrization \begin{align*} F:(0,\rho)\times N\to C(N)\cap B(0,\rho),\qquad F(r,y)=r y \end{align*} has induced metric \begin{align*} F^*g_{\mathbb R^n}=dr^2+r^2g_N. \end{align*} Therefore its $m$-dimensional volume element is \begin{align*} d\mathcal H^m_{C(N)}=r^{m-1}\,dr\,d\mathcal H^{m-1}_N. \end{align*} It follows that \begin{align*} \mathcal H^m(C(N)\cap B(0,\rho))=\int_N\int_0^\rho r^{m-1}\,dr\,d\mathcal H^{m-1}_N(y). \end{align*} Since \begin{align*} \int_0^\rho r^{m-1}\,dr=\frac{\rho^m}{m}, \end{align*} we obtain \begin{align*} \mathcal H^m(C(N)\cap B(0,\rho))=\frac{\rho^m}{m}\mathcal H^{m-1}(N). \end{align*} Substituting this into the definition of the density ratio gives \begin{align*} \Theta(C(N),0,\rho)=\frac{(\rho^m/m)\mathcal H^{m-1}(N)}{\theta_m\rho^m}. \end{align*} Cancelling the common factor $\rho^m$ gives \begin{align*} \Theta(C(N),0,\rho)=\frac{\mathcal H^{m-1}(N)}{m\theta_m}. \end{align*} Thus the density ratio is independent of $\rho$. Geometrically, this constancy reflects the dilation invariance of the cone, and the monotonicity defect vanishes away from the vertex. [/example] This example shows why cones are the equality models for monotonicity. The next section reverses the logic: if we zoom in at a point, monotonicity forces any limit to be conical. ## Blow-Up Sequences And Tangent Cones The basic local question near a possible singular point is what the submanifold looks like under magnification. Since the minimal surface equation is invariant under Euclidean dilations, rescaled minimal submanifolds remain stationary. The difficulty is compactness: after rescaling, curvature may grow, but the monotonicity formula supplies the area bounds needed for varifold convergence. [definition: Blow-Up Sequence] Let $\Sigma^m\subset \mathbb R^n$ be stationary in a neighbourhood of $x_0$. For $r>0$, define the dilation map \begin{align*} \eta_{x_0,r}:\mathbb R^n\to \mathbb R^n,\qquad \eta_{x_0,r}(x)=\frac{x-x_0}{r}. \end{align*} A blow-up sequence of $\Sigma$ at $x_0$ is a sequence of rescaled submanifolds or pushforward varifolds \begin{align*} \Sigma_j=(\eta_{x_0,r_j})_\#\Sigma \end{align*} with $r_j\downarrow 0$. [/definition] The rescaled object $\Sigma_j$ records geometry originally contained in $B(x_0,r_j R)$ as geometry inside $B(0,R)$. The density ratio is unchanged by this operation: \begin{align*} \Theta(\Sigma_j,0,R)=\Theta(\Sigma,x_0,r_jR). \end{align*} This motivates the next definition: if a blow-up sequence converges, the limit is the first-order geometric model of $\Sigma$ at $x_0$, and monotonicity will force that model to be conical. [definition: Tangent Cone] A stationary integral cone $C$ is a tangent cone to $\Sigma$ at $x_0$ if there exists a blow-up sequence $\Sigma_j=(\Sigma-x_0)/r_j$ such that $\Sigma_j$ converges to $C$ as integral varifolds on every compact subset of $\mathbb R^n$. [/definition] The word cone in this definition is part of the conclusion, not merely a naming convention. The theorem below is where the homogeneity enters. [quotetheorem:5673] [citeproof:5673] Tangent cones need not be unique in the generality stated here. The course will often use existence alone: any blow-up limit is a model for the singularity, and the density of the original surface is the density of that cone at its vertex. The compactness hypotheses also cannot be discarded. A sequence of rescalings has a varifold subsequential limit only because monotonicity gives uniform mass bounds on every fixed ball; without such bounds, mass can escape to infinity or accumulate with no Radon limit. Integrality is what prevents the limiting object from having arbitrary non-integer sheet density, and stationarity is what makes constant density ratios force conical invariance. The theorem therefore proves existence of stationary integral tangent cones, but it does not classify them, prove uniqueness, or imply smoothness of the original submanifold near the base point. [remark: Homogeneity And Links] If $C$ is a smooth cone away from the origin, then $C\cap S^{n-1}$ is its link. Stationarity of $C$ away from $0$ is equivalent to minimality of the link inside the unit sphere. Thus tangent cones translate local Euclidean singularity analysis into the study of minimal submanifolds of spheres. [/remark] The catenoid provides a geometric model for how blow-up can detect concentration at a neck rather than a smooth point. This example should be read as a warning about choosing scales: the same sequence can look flat at one magnification and highly curved at another. Monotonicity controls mass at all these scales, but it does not by itself identify which scale contains the most geometric information. Neck regions are where this distinction becomes visible, because their area may remain controlled while their curvature becomes large. [example: Blow-Up At A Neck Of A Catenoid Sequence] Let $a_j\downarrow 0$, and write the catenoid with neck radius $a_j$ as \begin{align*} X_j(u,v)=\bigl(a_j\cosh u\cos v,\ a_j\cosh u\sin v,\ a_j u\bigr), \qquad u\in\mathbb R,\ v\in[0,2\pi). \end{align*} At $u=0$ the waist circle has radius $a_j$ and lies in the plane $z=0$. The tangent vectors are \begin{align*} \partial_uX_j=\bigl(a_j\sinh u\cos v,\ a_j\sinh u\sin v,\ a_j\bigr). \end{align*} \begin{align*} \partial_vX_j=\bigl(-a_j\cosh u\sin v,\ a_j\cosh u\cos v,\ 0\bigr). \end{align*} Their metric coefficients are \begin{align*} |\partial_uX_j|^2=a_j^2\sinh^2u\cos^2v+a_j^2\sinh^2u\sin^2v+a_j^2=a_j^2\sinh^2u+a_j^2=a_j^2\cosh^2u. \end{align*} \begin{align*} |\partial_vX_j|^2=a_j^2\cosh^2u\sin^2v+a_j^2\cosh^2u\cos^2v=a_j^2\cosh^2u. \end{align*} \begin{align*} \partial_uX_j\cdot \partial_vX_j=-a_j^2\sinh u\cosh u\cos v\sin v+a_j^2\sinh u\cosh u\sin v\cos v+0=0. \end{align*} The standard catenoid has principal curvatures $\cosh^{-2}u$ and $-\cosh^{-2}u$, and dilation by $a_j$ multiplies principal curvatures by $a_j^{-1}$. Hence \begin{align*} \kappa_1(u)=\frac{1}{a_j\cosh^2u}. \end{align*} \begin{align*} \kappa_2(u)=-\frac{1}{a_j\cosh^2u}. \end{align*} Thus \begin{align*} H=\frac{\kappa_1+\kappa_2}{2}=\frac{1}{2}\left(\frac{1}{a_j\cosh^2u}-\frac{1}{a_j\cosh^2u}\right)=0. \end{align*} At the waist $u=0$, \begin{align*} |A|^2=\kappa_1^2+\kappa_2^2=\frac{1}{a_j^2}+\frac{1}{a_j^2}=\frac{2}{a_j^2}. \end{align*} So each surface is minimal, while its curvature becomes unbounded at the shrinking neck. For cylindrical radius $r>a_j$, the two sheets have heights \begin{align*} z_j^\pm(r)=\pm a_j\operatorname{arcosh}(r/a_j). \end{align*} Using $\operatorname{arcosh}s=\log(s+\sqrt{s^2-1})$ and $\sqrt{s^2-1}\le s$ for $s\ge 1$, we have \begin{align*} |z_j^\pm(r)|=a_j\log\left(\frac r{a_j}+\sqrt{\frac{r^2}{a_j^2}-1}\right). \end{align*} \begin{align*} |z_j^\pm(r)|\le a_j\log\left(\frac r{a_j}+\frac r{a_j}\right)=a_j\log\left(\frac{2r}{a_j}\right). \end{align*} For each fixed $r>0$, the product $a_j\log(2r/a_j)$ tends to $0$ as $a_j\downarrow 0$. Also \begin{align*} \frac{d z_j^\pm}{dr}=\pm a_j\frac{d}{dr}\operatorname{arcosh}(r/a_j)=\pm a_j\frac{1}{\sqrt{r^2-a_j^2}}. \end{align*} For each fixed $r>0$, this derivative tends to $0$. Therefore away from the origin the two graphical sheets converge to the plane $\{z=0\}$ with multiplicity $2$. If we blow up by the neck radius, then \begin{align*} \frac{1}{a_j}X_j(u,v)=\bigl(\cosh u\cos v,\ \cosh u\sin v,\ u\bigr), \end{align*} which is exactly the standard catenoid. Thus the neck-scale blow-up preserves the curved neck geometry. If instead $\lambda_j\downarrow 0$ and $a_j/\lambda_j\to 0$, then \begin{align*} \frac{1}{\lambda_j}X_j(u,v)=\left(\frac{a_j}{\lambda_j}\cosh u\cos v,\ \frac{a_j}{\lambda_j}\cosh u\sin v,\ \frac{a_j}{\lambda_j}u\right). \end{align*} This rescaled catenoid has neck radius $a_j/\lambda_j$, which tends to $0$. Applying the same height and slope estimates with $a_j$ replaced by $a_j/\lambda_j$ gives convergence away from the origin to the multiplicity-two plane. Thus the neck scale produces the standard catenoid, while any much larger vanishing scale produces the tangent cone, namely a double plane. [/example] The example is also a warning for compactness arguments. Area concentration, curvature concentration, and varifold convergence are related but not identical; choosing the scale determines which feature is visible. ## Local Area Bounds And Density Control Once monotonicity is available, the next problem is to turn pointwise density information into estimates on area in balls. These estimates are the basic compactness input for sequences of minimal submanifolds and later for energy measures of harmonic maps. [quotetheorem:5674] [citeproof:5674] This estimate is often used in reverse: a sequence with uniformly bounded mass in a slightly larger ball has uniformly bounded mass on all smaller balls, after rescaling or recentering. That is the hypothesis needed for varifold compactness. The estimate is not a substitute for stationarity. A nonstationary surface can place most of its area in a tiny inner ball while having no monotone density ratio relating that mass to the outer scale. The larger-ball mass control is also necessary: monotonicity propagates a finite bound from one scale to smaller scales, but it does not create a global bound from nothing. Even with these bounds, compactness is only varifold compactness; smooth convergence still requires additional regularity input such as density closeness, curvature estimates, or stability hypotheses. [remark: Density Gap At Smooth Points] For embedded smooth minimal submanifolds, density $1$ characterises multiplicity-one planar behaviour at a regular point. In regularity theory, quantitative versions say that if the density ratio is sufficiently close to $1$ on a ball, then the submanifold is graphical on a smaller ball with controlled estimates. This is the role of Allard regularity: it converts small excess and density closeness into a graphical description with quantitative control. [/remark] The next issue is whether a point missing from the surface is a genuine singularity or only an artefact of the domain. Monotonicity turns this into a question about finite density and stability. ## Removable Singularities And Concentration Suppose a minimal submanifold is defined in a punctured ball. The local question is whether the missing point can be filled in so that the result is still a minimal object. The answer depends on dimension, stability, and density: finite area prevents wild mass accumulation, while stability supplies curvature control away from the puncture. [quotetheorem:5675] [citeproof:5675] The density assumption is the point where the theorem excludes conical singularities and multiplicity. Stability alone controls oscillation and curvature in many dimensions, but it does not force the tangent cone to be a single plane. [example: Why Density Matters] Let $C(N)\subset \mathbb R^{m+1}$ be a nonflat minimal cone with isolated vertex at $0$, and write its link as \begin{align*} N=C(N)\cap S^m. \end{align*} For $0<\rho<R$, the cone parametrization \begin{align*} F:(0,\rho)\times N\to C(N)\cap B(0,\rho),\qquad F(r,y)=ry \end{align*} has induced metric \begin{align*} F^*g_{\mathbb R^{m+1}}=dr^2+r^2g_N, \end{align*} so the $m$-dimensional volume element is \begin{align*} d\mathcal H^m_{C(N)}=r^{m-1}\,dr\,d\mathcal H^{m-1}_N. \end{align*} Therefore \begin{align*} \mathcal H^m(C(N)\cap B(0,\rho))=\int_N\int_0^\rho r^{m-1}\,dr\,d\mathcal H^{m-1}_N(y). \end{align*} Since \begin{align*} \int_0^\rho r^{m-1}\,dr=\frac{\rho^m}{m}, \end{align*} we get \begin{align*} \mathcal H^m(C(N)\cap B(0,\rho))=\int_N \frac{\rho^m}{m}\,d\mathcal H^{m-1}_N(y). \end{align*} Thus \begin{align*} \mathcal H^m(C(N)\cap B(0,\rho))=\frac{\rho^m}{m}\mathcal H^{m-1}(N). \end{align*} Substituting this mass into the definition of the density ratio gives \begin{align*} \Theta(C(N),0,\rho)=\frac{\mathcal H^m(C(N)\cap B(0,\rho))}{\theta_m\rho^m}. \end{align*} Hence \begin{align*} \Theta(C(N),0,\rho)=\frac{(\rho^m/m)\mathcal H^{m-1}(N)}{\theta_m\rho^m}. \end{align*} Cancelling the common factor $\rho^m$ gives \begin{align*} \Theta(C(N),0,\rho)=\frac{\mathcal H^{m-1}(N)}{m\theta_m}. \end{align*} The right-hand side does not depend on $\rho$, so the density is finite and constant across scales. For a multiplicity-one flat $m$-plane, the link is an equatorial sphere $S^{m-1}\subset S^m$. Its cone is the plane itself, whose density ratio is $1$, so the preceding formula forces \begin{align*} \mathcal H^{m-1}(S^{m-1})=m\theta_m. \end{align*} Therefore, if the nonflat link satisfies \begin{align*} \mathcal H^{m-1}(N)>\mathcal H^{m-1}(S^{m-1})=m\theta_m, \end{align*} then \begin{align*} \Theta(C(N),0)=\frac{\mathcal H^{m-1}(N)}{m\theta_m}>1. \end{align*} The punctured cone $C(N)\setminus\{0\}$ is minimal away from $0$, because its link is minimal in the unit sphere. However, adding the vertex cannot make it a smooth embedded multiplicity-one hypersurface near $0$: a smooth embedded point has tangent plane density $1$, while this cone has density strictly larger than $1$. Thus finite monotone density alone does not guarantee a removable singularity; the density-one hypothesis rules out precisely this conical obstruction. [/example] The same monotonicity mechanism also describes concentration in sequences. If $\Sigma_j$ are minimal submanifolds with uniformly bounded area, then after passing to a subsequence their varifold limit may carry higher multiplicity or singular mass. Blow-up at points where the limiting density jumps identifies the local model for that concentration. [explanation: Concentration Analysis] In a compactness argument, the density ratio separates regular convergence from concentration. At points where the density stays close to $1$, regularity theorems usually promote varifold convergence to smooth graphical convergence on smaller balls. At points where density is larger, a blow-up sequence produces a tangent cone or bubble model that records the lost geometry. For minimal submanifolds, the concentration object is measured by $\mathcal H^m$ or, with multiplicity, by the varifold mass measure and density. In the harmonic map half of the course, the parallel object will be the energy measures $|\nabla u_j|^2\,d\mathcal L^m$ for a sequence of maps $u_j: \Omega\subset \mathbb R^m\to N$ with uniformly bounded energy; any weak limit may be accompanied by a defect measure recording energy lost in concentration. The same pattern appears: a monotone or almost monotone scale-invariant quantity yields blow-up limits, and the limits classify how compactness fails. [/explanation] This chapter therefore supplies the analytic language for singularities. Density ratios quantify local mass, monotonicity proves the [existence of densities](/theorems/4991), blow-ups extract limiting models, and tangent cones describe the first-order geometry of singular points. Monotonicity and tangent cones turn the qualitative variational picture into a quantitative local theory, showing how singular behavior can be measured and modeled. With that language in place, the next chapter uses curvature estimates and compactness to control how families of minimal surfaces can degenerate. # 5. Compactness And Curvature Estimates This chapter uses the second variation formula, the Jacobi operator, the monotonicity formula for minimal surfaces, and basic elliptic estimates for the minimal graph equation. It also assumes the standard local language of embedded surfaces in a Riemannian $3$-manifold, including the second fundamental form $A$, geodesic balls, injectivity radius, and smooth graphical convergence. With those tools in place, the chapter explains how compactness enters the analysis of minimal surfaces: stability gives pointwise curvature control, area and topology bounds limit where curvature can concentrate, and lamination limits describe what remains when smooth convergence fails at isolated points. ## Curvature Estimates For Stable Minimal Surfaces The first question is local: if a minimal surface is stable in a ball, how much can its second fundamental form grow near the centre? Stability is an integral inequality, while curvature is pointwise, so the main analytic work is to convert the stability inequality and the Simons-type equation for $|A|$ into a scale-invariant pointwise estimate. [definition: Stable Minimal Surface In A Ball] Let $(M^3,g)$ be a Riemannian $3$-manifold, let $B_M(p,r)$ be a geodesic ball, and let $\Sigma^2 \subset B_M(p,r)$ be a two-sided immersed minimal surface with unit normal $\nu$. The surface is stable in $B_M(p,r)$ if every $\phi \in C_c^\infty(\Sigma \cap B_M(p,r))$ satisfies \begin{align*} \int_\Sigma \left(|\nabla_\Sigma \phi|^2 - (|A|^2 + \operatorname{Ric}_M(\nu,\nu))\phi^2\right)\,d\mathcal H^2 \ge 0. \end{align*} [/definition] This is the same stability inequality obtained from second variation, now restricted to compactly supported variations inside the ball. The compact support condition is important because the estimate is local and should not depend on boundary behaviour far away. [quotetheorem:5676] [citeproof:5676] The estimate is scale invariant: after rescaling the metric by $r^{-2}$, the right-hand side becomes a uniform constant. It says that stable sheets have no necks, bubbles, or sharp folds at scales where the ambient geometry is controlled. Each hypothesis has a distinct role. The ambient curvature and injectivity-radius assumptions ensure that after rescaling small balls look uniformly Euclidean, so the contradiction sequence has a Euclidean blow-up limit rather than a limit distorted by collapsing ambient geometry. Proper embeddedness and the boundary condition keep the interior ball from seeing artificial boundary sheets, while two-sidedness gives a globally defined normal and hence the scalar stability inequality used in the argument. Stability is the decisive analytic input: without it, the catenoid shows that curvature can concentrate near a shrinking neck while the surface remains minimal. The theorem also does not assert global compactness or rule out curvature blow-up at the boundary; it is an interior estimate at a controlled scale, and this local nature is what makes it useful inside the compactness arguments that follow. [example: Graphical Minimal Disks] Let $u_k:B(0,1)\subset\mathbb R^2\to\mathbb R$ solve the minimal graph equation \begin{align*} \operatorname{div}\left(\frac{\nabla u_k}{\sqrt{1+|\nabla u_k|^2}}\right)=0, \end{align*} with $|u_k|\le C_0$ and $|\nabla u_k(0)|\le C_1$, and let \begin{align*} F_k(x_1,x_2)=(x_1,x_2,u_k(x_1,x_2)). \end{align*} The tangent vectors are \begin{align*} \partial_1F_k=(1,0,\partial_1u_k),\qquad \partial_2F_k=(0,1,\partial_2u_k), \end{align*} so the induced metric is \begin{align*} (g_k)_{ij} =\partial_iF_k\cdot \partial_jF_k =\delta_{ij}+(\partial_i u_k)(\partial_j u_k). \end{align*} The upward unit normal is \begin{align*} \nu_k=\frac{(-\partial_1u_k,-\partial_2u_k,1)}{\sqrt{1+|\nabla u_k|^2}}, \end{align*} and the scalar mean curvature of the graph is \begin{align*} H_k =\operatorname{div}\left(\frac{\nabla u_k}{\sqrt{1+|\nabla u_k|^2}}\right) =0. \end{align*} Thus each $\Sigma_k=F_k(B(0,1))$ is a minimal disk. Since minimal graphs minimize area among compactly supported graphical variations, their second variation is nonnegative, so the disks are stable. Fix a compact set $K\subset B(0,1/2)$. Choose $\rho>0$ so that the Euclidean $\rho$-neighbourhood of $K$ in $\mathbb R^2$ is contained in $B(0,1)$. For every $x\in K$, the portion of $\Sigma_k$ over $B_{\mathbb R^2}(x,\rho)$ is a stable minimal surface in a Euclidean ball of comparable radius, and the ambient curvature and injectivity-radius constants are those of $\mathbb R^3$. Applying *[Schoen Curvature Estimate For Stable Minimal Surfaces](/theorems/5676)* at this fixed scale gives \begin{align*} \sup_{F_k(K)} |A_{\Sigma_k}|^2 \le C_K, \end{align*} where $C_K$ is independent of $k$. Interior gradient estimates for the minimal graph equation, using the height bound $|u_k|\le C_0$, also give \begin{align*} \sup_K |\nabla u_k|\le C'_K. \end{align*} In graph coordinates, \begin{align*} (h_k)_{ij} =\partial_i\partial_jF_k\cdot \nu_k =\frac{\partial_{ij}u_k}{\sqrt{1+|\nabla u_k|^2}}, \end{align*} so the bounds for $|A_{\Sigma_k}|$ and $|\nabla u_k|$ give uniform bounds for the Hessians $\partial_{ij}u_k$ on $K$. With the gradient bound, the equation is uniformly elliptic on $K$, and Schauder estimates then bound all higher derivatives of $u_k$ on smaller compact subsets. Arzela-Ascoli therefore gives a subsequence converging in $C^\infty_{\mathrm{loc}}(B(0,1/2))$ to a function $u_\infty$, and passing to the limit in the displayed minimal graph equation shows that $u_\infty$ is again a minimal graph. [/example] This example is the model case behind the general estimate: local stability supplies curvature control, and curvature control converts the nonlinear equation into a compact elliptic problem. The next step is to ask when a whole sequence of closed or properly embedded minimal surfaces has enough local stability, or enough replacement control, to converge. ## Compactness From Area And Topology Bounds The second question is global-to-local: which coarse geometric bounds prevent uncontrolled curvature concentration across a sequence of embedded minimal surfaces? The theorem below is a Choi-Schoen type compactness principle, in the later area-and-topology formulation used in minimal-surface compactness theory. It should be read as a compactness package built from curvature estimates, removable singularities, and local structure theory for embedded minimal surfaces, rather than as the narrower original positive-Ricci Choi-Schoen statement. [definition: Smooth Local Convergence Of Surfaces] Let $\Sigma_k$ and $\Sigma$ be embedded surfaces in a Riemannian $3$-manifold $M$. The sequence $\Sigma_k$ converges smoothly locally to $\Sigma$ on an open set $U\subset M$ if for every $q\in \Sigma\cap U$ there is a coordinate chart and a domain $\Omega\subset T_q\Sigma$ such that, for all large $k$, $\Sigma_k$ is a finite union of graphs of smooth maps $u_{k,j}:\Omega\to\mathbb R$ over $\Omega$, and $u_{k,j}\to 0$ in $C^m(K)$ for every compact $K\subset\Omega$ and every $m\ge 1$. [/definition] This definition records convergence with multiplicity by allowing several local graphs. Multiplicity is unavoidable: a sequence can approach the same limiting sheet from both sides even when every member of the sequence is embedded. The compactness problem is to identify hypotheses that force this local graphical description to hold after discarding only a controlled singular set. In the Choi-Schoen compactness picture used here, the conclusion is that a subsequence converges smoothly locally away from a finite curvature-concentration set $\mathcal S$. The limiting object is a smooth embedded minimal surface or lamination on $M\setminus\mathcal S$, and removable-singularity input is then used to decide which punctures can be filled smoothly. The finite set $\mathcal S$ consists of possible curvature concentration points. The area and genus bounds imply that only finitely many disjoint concentration regions can occur, because each such region consumes a definite amount of topology or area at a definite scale. The hypotheses are not interchangeable. Closedness of the ambient manifold supplies a finite covering by uniformly controlled coordinate balls; on a noncompact ambient space, surfaces can escape to infinity or see degenerating geometry unless extra local bounds are imposed. Closedness of the surfaces removes boundary degeneration, while embeddedness gives the local sheet and neck structure used to turn curvature bounds into graphical convergence; immersed examples can have self-intersections or accumulating sheets that do not fit this conclusion. The area bound controls the number of sheets visible in a fixed region, as sequences with increasing area in a flat torus can pass through the same ball with unbounded multiplicity. The genus bound controls topology concentration: without it, handles can pile up at more and more scales, producing infinitely many neck regions even when the surfaces are individually smooth. Thus the theorem is not merely an Arzela-Ascoli statement for parametrisations; it is an ambient compactness result designed to isolate finitely many geometric failures before lamination theory takes over. [example: Failure Without Area Control] In a flat three-torus $T^3=\mathbb R^3/\mathbb Z^3$, consider embedded closed minimal surfaces $\Sigma_k$ with increasing genus and area, arranged so that in some coordinate ball $U\cong B_{\mathbb R^3}(0,2r)$ each $\Sigma_k$ contains $N_k\to\infty$ disjoint graphical sheets over the same disk $D=B_{\mathbb R^2}(0,r)$. Write these sheets as \begin{align*} \Gamma_{k,j}=\{(x_1,x_2,u_{k,j}(x_1,x_2)):(x_1,x_2)\in D\}, \qquad 1\le j\le N_k . \end{align*} For one sheet, the induced area element is \begin{align*} d\mathcal H^2_{\Gamma_{k,j}} = \sqrt{1+|\nabla u_{k,j}|^2}\,dx_1dx_2, \end{align*} so \begin{align*} \mathcal H^2(\Gamma_{k,j}) = \int_D \sqrt{1+|\nabla u_{k,j}|^2}\,dx_1dx_2 \ge \int_D 1\,dx_1dx_2 = \pi r^2. \end{align*} Because the sheets are disjoint subsets of $\Sigma_k\cap U$, \begin{align*} \mathcal H^2(\Sigma_k\cap U) \ge \sum_{j=1}^{N_k}\mathcal H^2(\Gamma_{k,j}) \ge \sum_{j=1}^{N_k}\pi r^2 = N_k\pi r^2 \longrightarrow \infty. \end{align*} If a subsequence converged smoothly locally on $U$ with finite multiplicity $m$, then on the smaller ball the surfaces would be the union of at most $m$ smooth graphical sheets for all large $k$, giving a uniform local area bound. The displayed lower bound contradicts this once $N_k>m$. Thus smoothness and closedness of each individual surface do not prevent uncontrolled local multiplicity; the area and topology assumptions in the [compactness theorem](/theorems/2748) are what rule out this kind of degeneration. [/example] The compactness theorem gives smooth convergence away from finitely many points, but it does not by itself describe the geometry near those points. To understand the approach to singularity formation, the limiting language must be broadened from single surfaces to laminations. ## Lamination Limits And Removable Singularities The third question asks what the eye sees before a singularity forms. Near a neck-collapse point, the sequence may resemble many almost-flat sheets connected by a tiny bridge; away from the bridge the limiting object is not a single embedded surface with a chosen multiplicity, but a collection of disjoint leaves filling part of the ambient space. [definition: Minimal Lamination] Let $U$ be an open subset of a Riemannian $3$-manifold $(M^3,g)$. A minimal lamination of $U$ is a closed subset $\mathcal L\subset U$ together with a decomposition into connected smooth embedded minimal surfaces called leaves, such that every point $p\in\mathcal L$ has a coordinate neighbourhood $V$ in which \begin{align*} \mathcal L\cap V = \bigcup_{t\in T} \{(x_1,x_2,x_3): x_3 = f_t(x_1,x_2) \} \end{align*} for a closed parameter set $T\subset\mathbb R$, a domain $\Omega\subset\mathbb R^2$, and maps $f_t:\Omega\to\mathbb R$ with $f_t\in C^\infty(\Omega;\mathbb R)$, whose graphs are pairwise disjoint. [/definition] A lamination is weaker than a foliation because it need not fill the whole neighbourhood. It is stronger than a varifold limit because it preserves the local graphical leaf structure away from the singular set. [quotetheorem:5678] [citeproof:5678] This theorem is the analytic mechanism that converts punctured convergence into a genuine limiting object. The hypotheses are tailored to the estimates produced in compactness arguments: curvature may grow like the inverse square of the distance to the puncture, but not faster. The scale-invariant curvature hypothesis is essential because an isolated puncture can otherwise hide faster-than-quadratic bending or spiralling. A punctured minimal graph with uncontrolled gradient near the missing point, or a lamination modelled on sheets that spiral into an axis, may fail to admit a smooth leaf through the puncture even though every compact subset away from the puncture looks regular. The theorem also does not say that the original sequence converges smoothly at the puncture: in a neck-collapse sequence, curvature can still blow up at the shrinking neck. Its conclusion is instead about the limiting lamination, and this is exactly the forward link to compactness theory: once a sequence has produced a punctured lamination away from finitely many bad points, the removable theorem identifies which of those points are artifacts of the limiting description rather than genuine singularities of the limit. [example: Catenoid Neck Collapse] For $a>0$, write the catenoid of waist radius $a$ as \begin{align*} X_a(t,\theta)=(a\cosh t\cos\theta,\ a\cosh t\sin\theta,\ at), \end{align*} where $t\in\mathbb R$ and $\theta\in[0,2\pi)$. Its waist is the circle $t=0$, whose radius is $a$, so the waist shrinks to the origin as $a\to 0$. If $\rho=\sqrt{x_1^2+x_2^2}$, then $\rho=a\cosh t$. Hence, on the region $\rho>a$, the two halves of the catenoid are graphs over the punctured $x_1x_2$-plane with heights \begin{align*} x_3=\pm a\,\operatorname{arcosh}(\rho/a). \end{align*} For $\rho\ge \delta>0$ and $a<\delta/2$, \begin{align*} \left|a\,\operatorname{arcosh}(\rho/a)\right|=a\log\left(\frac{\rho}{a}+\sqrt{\frac{\rho^2}{a^2}-1}\right). \end{align*} Since $\sqrt{\rho^2/a^2-1}\le \rho/a$, this gives \begin{align*} \left|a\,\operatorname{arcosh}(\rho/a)\right|\le a\log\left(\frac{2\rho}{a}\right). \end{align*} On every compact annulus $\delta\le \rho\le R$, the right-hand side is bounded above by $a\log(2R/a)$, which tends to $0$ as $a\to 0$. Now set \begin{align*} f_a(\rho)=a\,\operatorname{arcosh}(\rho/a). \end{align*} Using $(\operatorname{arcosh} s)'=(s^2-1)^{-1/2}$ for $s>1$, we get \begin{align*} f_a'(\rho)=a\cdot \frac{1}{\sqrt{(\rho/a)^2-1}}\cdot \frac{1}{a}. \end{align*} Since $\sqrt{(\rho/a)^2-1}=\sqrt{\rho^2-a^2}/a$, this becomes \begin{align*} f_a'(\rho)=\frac{a}{\sqrt{\rho^2-a^2}}. \end{align*} If $a<\rho/2$, then $\rho^2-a^2\ge 3\rho^2/4$, so \begin{align*} |f_a'(\rho)|\le \frac{2a}{\rho}. \end{align*} Differentiating the last displayed formula gives \begin{align*} f_a''(\rho)=-a\rho(\rho^2-a^2)^{-3/2}. \end{align*} On each annulus $\delta\le \rho\le R$, this is bounded in absolute value by a constant times $a$, and the same differentiation pattern gives convergence of every higher derivative to $0$ there. Thus the upper and lower sheets converge smoothly on compact subsets of $\{x_3=0\}\setminus\{0\}$, so the limiting lamination away from the origin is the punctured plane with multiplicity two. The curvature concentrates at the shrinking waist. The tangent vectors are \begin{align*} X_t=(a\sinh t\cos\theta,\ a\sinh t\sin\theta,\ a). \end{align*} \begin{align*} X_\theta=(-a\cosh t\sin\theta,\ a\cosh t\cos\theta,\ 0). \end{align*} Therefore the induced metric coefficients are \begin{align*} E=X_t\cdot X_t=a^2\cosh^2 t. \end{align*} \begin{align*} F=X_t\cdot X_\theta=0. \end{align*} \begin{align*} G=X_\theta\cdot X_\theta=a^2\cosh^2 t. \end{align*} With the unit normal \begin{align*} \nu=\frac{(-\cos\theta,\ -\sin\theta,\ \sinh t)}{\cosh t}, \end{align*} the second derivatives are \begin{align*} X_{tt}=(a\cosh t\cos\theta,\ a\cosh t\sin\theta,\ 0). \end{align*} \begin{align*} X_{t\theta}=(-a\sinh t\sin\theta,\ a\sinh t\cos\theta,\ 0). \end{align*} \begin{align*} X_{\theta\theta}=(-a\cosh t\cos\theta,\ -a\cosh t\sin\theta,\ 0). \end{align*} Taking dot products with $\nu$ gives \begin{align*} h_{tt}=X_{tt}\cdot \nu=-a. \end{align*} \begin{align*} h_{t\theta}=X_{t\theta}\cdot \nu=0. \end{align*} \begin{align*} h_{\theta\theta}=X_{\theta\theta}\cdot \nu=a. \end{align*} Since $g^{tt}=g^{\theta\theta}=1/(a^2\cosh^2 t)$ and $g^{t\theta}=0$, the principal curvatures are \begin{align*} \kappa_1=-\frac{1}{a\cosh^2 t}. \end{align*} \begin{align*} \kappa_2=\frac{1}{a\cosh^2 t}. \end{align*} Hence \begin{align*} |A|^2=\kappa_1^2+\kappa_2^2=\frac{2}{a^2\cosh^4 t}. \end{align*} At the waist $t=0$, this becomes \begin{align*} |A|^2=\frac{2}{a^2}\longrightarrow \infty. \end{align*} Thus the missing point in the limiting punctured plane is removable at the level of the lamination limit by the *Removable Singularity For Minimal Laminations*, but the catenoids themselves do not converge smoothly through the origin because their curvatures become unbounded there. [/example] This is the local picture behind many degenerations: smooth convergence persists on the complement of a small bad set, while topology disappears into regions where curvature concentrates. Compactness theorems separate these two phenomena so that estimates can be proved away from the bad set and singularity-removal arguments can analyse the limiting object. [remark: Multiplicity And Stability Of Limit Leaves] When convergence to a leaf occurs with multiplicity greater than one, the separation between adjacent sheets often produces a positive solution of the Jacobi equation on the limiting leaf. Such a solution implies stability of the leaf, linking multiplicity in compactness theory back to the stability estimates at the start of the chapter. [/remark] The chapter's estimates form the bridge between local elliptic regularity and global moduli-space compactness. Stability gives pointwise curvature bounds, area and genus bounds limit the number of concentration points, and lamination theory records the geometric limit after necks or bubbles have been removed from the smooth part of the convergence. Curvature bounds and compactness extend the local blow-up picture into a global convergence theory for minimal surfaces. Once those tools are in place, the course can return to the original spanning problem of Plateau, now equipped with the analytic machinery needed to treat existence and regularity. # 6. Plateau-Type Problems And Regularity Plateau's problem asks for a surface of least area among all surfaces spanning a fixed boundary. This chapter assumes the earlier material on first variation from Chapter 1, stability from Chapter 2, monotonicity and blow-up from Chapter 4, Sobolev compactness, and the basic language of currents and varifolds. Earlier chapters treated smooth critical points, stability, compactness, and blow-up; the present chapter explains how the variational problem is solved when smooth parametrisations are too rigid. The central move is to enlarge the class of competitors to integral currents, prove existence by compactness and lower semicontinuity, and then recover smooth geometry away from a controlled singular set. ## The Boundary Spanning Problem For Currents The first issue in Plateau's problem is not regularity but compactness: a minimizing sequence of smooth spanning surfaces can develop necks, folds, or changes of topology. Integral currents provide a weak category in which oriented surfaces, multiplicities, and boundaries all have stable limiting behaviour. [definition: Integral Current With Prescribed Boundary] Let $U$ be an open subset of $\mathbb R^n$, let $1 \le m \le n$, and let $\Gamma$ be an integral $(m-1)$-current in $U$ with $\partial \Gamma = 0$. The admissible class with boundary $\Gamma$ is \begin{align*} \mathcal A_\Gamma(U) := \{T \in \mathcal I_m(U) : \partial T = \Gamma\}. \end{align*} An integral current $T \in \mathcal I_m(U)$ has prescribed boundary $\Gamma$ if \begin{align*} \partial T = \Gamma \end{align*} as currents in $U$. [/definition] This definition separates the geometric surface from a particular parametrisation. To turn the spanning condition into a variational problem, we also need the weak notion of area that is lower semicontinuous under current convergence. [definition: Mass Of A Current] Let $U \subset \mathbb R^n$ be open, and let $\mathcal I_m(U)$ denote the class of integral $m$-currents in $U$. The mass functional is the map \begin{align*} \mathbf M : \mathcal I_m(U) \to [0,\infty] \end{align*} defined by \begin{align*} \mathbf M(T) := \|T\|(U) \end{align*} for each $T \in \mathcal I_m(U)$. [/definition] For a smooth oriented embedded $m$-submanifold $M \subset U$ with multiplicity one, this mass is its $\mathcal H^m$-area. The next step is to formulate least area among all currents satisfying the same boundary equation. [definition: Area-Minimizing Current With Boundary] Let $\Gamma$ be an integral $(m-1)$-current in $U \subset \mathbb R^n$. An integral $m$-current $T$ is area-minimizing with boundary $\Gamma$ in $U$ if $\partial T = \Gamma$ and \begin{align*} \mathbf M(T) \le \mathbf M(S) \end{align*} for every integral $m$-current $S$ in $U$ with $\partial S = \Gamma$. [/definition] The phrase "area-minimizing" is stronger than stationary or stable. Once the admissible class and functional are fixed, the direct method asks whether the infimum is achieved by an actual current rather than only approached by a sequence. [quotetheorem:5679] [citeproof:5679] This is the direct-method existence result for currents: compactness supplies a weak limit, closure preserves the constraint, and lower semicontinuity prevents loss of area in the limit. The nonemptiness and finite-mass hypotheses are part of the mechanism, not bookkeeping: without at least one competitor the infimum is taken over an empty class, and without a finite-mass competitor there is no bounded minimizing sequence to which compactness can be applied. The compact support hypothesis is also essential rather than cosmetic. Without some tightness condition, a minimizing sequence in a noncompact ambient space can drift farther and farther away while keeping the same boundary behaviour on compact sets, so the local current limit need not retain the intended global spanning surface. Equivalently, the theorem proves existence for the compactly supported class $\mathcal A_{\Gamma,K}(U)$; solving the unrestricted problem requires an additional argument that minimizing sequences cannot escape every compact subset of $U$. The theorem also does not claim uniqueness or boundary smoothness. Two different currents can attain the same minimum, and even when an interior regularity theorem applies, the behaviour near $\operatorname{spt}\Gamma$ depends on extra hypotheses on the boundary. These limitations explain the next two discussions: first nonuniqueness, then the separation between boundary and interior regularity. [example: Soap Film Spanning A Wire Curve] Let $\Gamma$ be the integral $1$-current induced by a smooth embedded closed curve in $\mathbb R^3$, with the chosen orientation followed once around the wire. Fix a compact set $K \subset \mathbb R^3$ containing $\operatorname{spt}\Gamma$, and consider the restricted admissible class \begin{align*} \mathcal A_{\Gamma,K}(\mathbb R^3) = \{R \in \mathcal I_2(\mathbb R^3) : \partial R=\Gamma,\ \operatorname{spt}R\subset K\}. \end{align*} If this class is nonempty and contains a finite-mass competitor, *Existence Of Area-Minimizing Currents With Prescribed Boundary* gives a current $T\in \mathcal A_{\Gamma,K}(\mathbb R^3)$ satisfying \begin{align*} \partial T = \Gamma. \end{align*} It also satisfies \begin{align*} \operatorname{spt}T \subset K. \end{align*} Finally, it attains the minimum \begin{align*} \mathbf M(T) = \inf\{\mathbf M(R):R\in \mathcal A_{\Gamma,K}(\mathbb R^3)\}. \end{align*} Thus $T$ minimizes area among oriented integral $2$-currents spanning the wire and remaining inside the chosen compact set. For the unrestricted Plateau problem, the admissible class is instead \begin{align*} \mathcal A_{\Gamma}(\mathbb R^3) = \{R \in \mathcal I_2(\mathbb R^3):\partial R=\Gamma\}, \end{align*} with no support condition. The compactness theorem used in the direct method applies to a minimizing sequence only after one knows that its supports do not escape to infinity, so an additional tightness or convex-hull argument is needed to reduce the unrestricted problem to a compactly supported one. If such a minimizer $T$ is obtained and, away from the wire, $T=q[M]$ for a smooth embedded surface $M$ with integer multiplicity $q\ge 1$, then every compactly supported interior variation field $X$ gives \begin{align*} \delta \mathbf M(q[M])(X)=0 \end{align*} by stationarity of an interior area minimizer. The smooth first variation formula gives \begin{align*} \delta \mathbf M(q[M])(X)=-q\int_M H_M\cdot X\,d\mathcal H^2. \end{align*} Combining the two identities gives \begin{align*} \int_M H_M\cdot X\,d\mathcal H^2=0 \end{align*} for every compactly supported interior variation field $X$, since $q>0$. The fundamental lemma for smooth vector fields then gives $H_M=0$ in the interior. Hence the weak current minimizer recovers the classical minimal surface equation at every smooth interior point. [/example] Physical soap films are not always oriented currents, because real films may meet in triple junctions. The current model is therefore not the whole physical Plateau problem; it is the oriented area-minimizing version, which is the natural setting for the direct method and the regularity theorems in this chapter. ## Nonuniqueness And The Meaning Of A Minimizer Existence does not imply uniqueness. The boundary may admit several distinct minimizers with the same area, and this is one reason the weak formulation is framed as an existence theorem rather than as a construction of a canonical surface. [example: Nonunique Spanning Surfaces] Let $\Gamma$ be an integral $1$-current in $\mathbb R^3$ that is invariant under an isometry $\sigma$ with $\sigma_\#\Gamma=\Gamma$, and suppose there are two geometrically distinct oriented spanning disks represented by integral $2$-currents $T_1$ and $T_2$ such that \begin{align*} \partial T_1=\Gamma, \qquad \partial T_2=\Gamma, \qquad T_2=\sigma_\# T_1, \qquad T_1\ne T_2. \end{align*} Because $\sigma$ is an isometry, it preserves $2$-dimensional area and hence preserves mass of integral $2$-currents: \begin{align*} \mathbf M(T_2)=\mathbf M(\sigma_\#T_1)=\mathbf M(T_1). \end{align*} If $T_1$ is minimizing in the current class with boundary $\Gamma$, then for every $S\in \mathcal I_2(\mathbb R^3)$ with $\partial S=\Gamma$, \begin{align*} \mathbf M(T_1)\le \mathbf M(S). \end{align*} Using $\mathbf M(T_2)=\mathbf M(T_1)$, the same inequality becomes \begin{align*} \mathbf M(T_2)=\mathbf M(T_1)\le \mathbf M(S), \end{align*} so $T_2$ is also minimizing among currents with boundary $\Gamma$. Thus the same boundary has at least two distinct global minimizers, namely $T_1$ and $T_2$. This illustrates that existence of an area-minimizing current does not select a canonical spanning surface: symmetries or other geometric features of the boundary can make two different currents attain the same minimum mass. [/example] Nonuniqueness also affects how regularity is used. A regularity theorem describes each minimizer after it has been found; it does not select between different minimizers. [remark: Boundary Versus Interior Questions] The direct method gives a current with the required boundary, but the boundary itself is a separate regularity problem. Interior regularity asks what happens at points of $\operatorname{spt} T \setminus \operatorname{spt} \partial T$. [Boundary regularity](/theorems/99) needs compatibility conditions on $\Gamma$ and more refined barrier or reflection arguments. [/remark] This chapter concentrates on interior points because the key regularity mechanisms already appear there. Boundary regularity belongs to the more delicate part of Plateau theory, especially when the boundary curve or ambient constraint has limited smoothness. ## Allard Regularity And Small Excess The next problem is to recognise when a weak minimal surface is actually a smooth surface. The answer is local and quantitative: if the current or varifold is close enough to a plane at one scale and has sufficiently small first variation, then it is represented by a smooth graph at a smaller scale. [definition: Density Of A Rectifiable Current] Let $T$ be an $m$-dimensional rectifiable current in $U \subset \mathbb R^n$. The density assignment is the partially defined map \begin{align*} \Theta^m(\|T\|,\cdot) : U \to [0,\infty] \end{align*} given, at points where the limit exists, by sending $x_0$ to \begin{align*} \Theta^m(\|T\|, x_0) := \lim_{r \downarrow 0} \frac{\|T\|(B(x_0,r))}{\omega_m r^m}, \end{align*} where $\omega_m = \mathcal L^m(B(0,1))$. [/definition] For area-minimizing currents, the monotonicity formula gives existence of the density at interior points. Density alone, however, does not record the direction in which the surface is approaching its tangent object: two sheets can have nearly the same mass ratio in a ball while oscillating between different tangent directions at smaller scales. To pass from density information to a graphical representation, we need a quantitative measure of how far the tangent planes tilt away from a fixed plane. [definition: Cylindrical Excess] Let $\mathcal R_m(\mathbb R^n)$ denote the class of $m$-dimensional rectifiable currents in $\mathbb R^n$, and let $G(m,n)$ denote the Grassmannian of unoriented $m$-planes in $\mathbb R^n$. The excess functional is the map \begin{align*} E : \mathcal R_m(\mathbb R^n) \times G(m,n) \times \mathbb R^n \times (0,\infty) \to [0,\infty] \end{align*} defined by \begin{align*} E(T,P,x_0,r) := r^{-m}\int_{B(x_0,r)} |\pi_{T_xT} - \pi_P|^2\,d\|T\|(x), \end{align*} where $\pi_{T_xT}$ is the approximate tangent-plane projection and $\pi_P$ is the [orthogonal projection](/theorems/437) onto $P$. [/definition] Small excess says that the tangent planes of the weak surface are close in average to a fixed plane. This is the hypothesis that turns a mass-ratio statement into geometric flatness: it specifies not only how much area lies in a ball, but also which plane the varifold resembles. The key regularity question is whether sufficiently small excess and mean curvature force the support to be a classical graph. Allard's theorem is the local bridge from weak varifold data to a graphical surface. The hypotheses below include both a density bound and explicit closeness to a plane; this plane-closeness is what allows the proof to begin with height and tilt estimates rather than merely with a scalar mass ratio. [quotetheorem:5680] [citeproof:5680] For area-minimizing currents the generalized mean curvature vanishes in the interior, so Allard's theorem applies once a tangent cone is a multiplicity-one plane and the excess is small at some scale. Each hypothesis has a specific role. The unit-density condition rules out higher multiplicity planes, the mass-ratio bound prevents hidden area concentration, the tilt-excess assumption excludes tangent planes that point in the wrong directions, the height-excess assumption excludes sheets that stay far from the reference cylinder even when their tangents are often well aligned, and the $L^p$ smallness of $H$ prevents a nearly flat surface from bending too violently between scales. The theorem does not classify all points of a minimizer. A multiplicity-two plane has the same underlying support as a plane but density $2$, so it violates the unit-density hypothesis and cannot be concluded to be a single sheet. A nonflat minimizing cone, such as the Simons cone in its critical dimension, has scale-invariant mass behaviour but fails the plane-excess hypothesis. A pair of nearly parallel sheets separated in the normal direction can have small tilt relative to $P$ while failing a small height bound in the chosen cylinder. These obstructions explain why the next section studies the singular set: Allard identifies regular points once flatness is known, while blow-up and dimension-reduction arguments explain how often flatness can fail. [example: Plane As A Regular Tangent Cone] Let $x_0 \notin \operatorname{spt}\partial T$, and define the rescaled currents \begin{align*} T_r := (\eta_{x_0,r})_\# T, \qquad \eta_{x_0,r}(x) := \frac{x-x_0}{r}. \end{align*} Suppose $T_{r_j}$ converges as $r_j \downarrow 0$ to the multiplicity-one plane current $[P]$. For the unit ball, the mass of the limit is \begin{align*} \mathbf M([P]\llcorner B(0,1)) = \mathcal H^m(P\cap B(0,1)) = \omega_m. \end{align*} By the scaling of $m$-dimensional mass under $\eta_{x_0,r}$, \begin{align*} \mathbf M(T_r\llcorner B(0,1)) = r^{-m}\mathbf M(T\llcorner B(x_0,r)). \end{align*} Hence along the blow-up sequence, \begin{align*} r_j^{-m}\mathbf M(T\llcorner B(x_0,r_j)) = \mathbf M(T_{r_j}\llcorner B(0,1)) \longrightarrow \omega_m. \end{align*} The monotonicity formula for interior area minimizers gives the existence of \begin{align*} \Theta^m(\|T\|,x_0) = \lim_{r\downarrow 0} \frac{\|T\|(B(x_0,r))}{\omega_m r^m}, \end{align*} so the subsequential limit above forces \begin{align*} \Theta^m(\|T\|,x_0)=1. \end{align*} Interior area-minimality also implies stationarity under compactly supported variations in $B(x_0,r)$ whenever $B(x_0,r)\cap \operatorname{spt}\partial T=\varnothing$, so the generalized mean curvature satisfies \begin{align*} H=0 \end{align*} there. Since $T_{r_j}\to [P]$, the height excess over $P$ and the tilt excess relative to $P$ tend to $0$ on fixed balls after rescaling. Therefore, for all sufficiently small scales, the density, mass-ratio, height-excess, tilt-excess, and mean-curvature hypotheses in *Allard Regularity Theorem* are satisfied. It follows that, after rotating coordinates so that $P=\mathbb R^m\times\{0\}$, there is a function \begin{align*} u\in C^{1,\alpha}(B_P(x_0,\theta r);\mathbb R^{n-m}) \end{align*} such that \begin{align*} \operatorname{spt}T\cap \bigl(B_P(x_0,\theta r)\times P^\perp\bigr) = \{y+u(y):y\in B_P(x_0,\theta r)\}. \end{align*} On this graphical region, the weak-to-classical principle gives zero mean curvature. Thus $u$ satisfies the minimal surface system, and the standard elliptic bootstrapping for that system upgrades the graph from $C^{1,\alpha}$ to smooth wherever the graphical equation is valid. The multiplicity-one planar blow-up therefore identifies $x_0$ as a regular interior point of the current. [/example] The hypothesis that the tangent cone be a single plane is a genuine geometric restriction. Singularities are precisely the places where no such graphical scale can be found, or where the limiting cone has higher multiplicity or nonflat geometry. ## Singular Sets And Dimension Restrictions A weak minimizer can be smooth on most of its support and still possess singular points. The regularity problem therefore asks for the size and structure of the singular set, not merely whether it is empty. [definition: Regular And Singular Points Of An Area Minimizer] Let $T$ be an area-minimizing integral $m$-current in $U \subset \mathbb R^n$. An interior point $x_0 \in \operatorname{spt} T \setminus \operatorname{spt}\partial T$ is regular if, in a neighbourhood of $x_0$, $\operatorname{spt} T$ is a smooth embedded $m$-submanifold with constant integer multiplicity. The singular set $\operatorname{sing} T$ is the set of interior points that are not regular. [/definition] This definition is local in the support, so it ignores boundary irregularities and focuses on failures of smooth embeddedness in the interior. The obstruction is that an area minimizer may have tangent cones that are themselves minimizing but not flat, and such cones can persist under blow-up at a singular point. Regularity theory therefore has to prove not only that singular points are exceptional, but that the exceptional set is quantitatively small enough to be harmless in variational arguments. [quotetheorem:5681] [citeproof:5681] The theorem explains why low-dimensional Plateau problems often have smooth solutions, while higher-dimensional problems require singular models. The hypotheses cannot be replaced by mere stationarity: stable or stationary varifolds may have much larger singular sets, because they need not minimize against all compactly supported competitors. The codimension-one assumption is also responsible for the sharper $m-7$ bound. The threshold is tied to the cone classification: the Simons cone is a codimension-one area-minimizing cone with an isolated singularity in dimension $m=7$, so the estimate cannot be improved in general beyond $\dim_{\mathcal H}(\operatorname{sing}T)\le m-7$. In higher codimension, branch points and other nonflat minimizing cone phenomena can occur in lower-dimensional models, so the general theory only gives the broader $m-2$ bound without extra structure. The result does not describe the singular set point by point, nor does it give boundary regularity. It gives a size bound strong enough to make singularities negligible for many variational arguments, while leaving the local classification of singular cones as a separate problem. The Simons cone example shows the sharp obstruction behind the dimension threshold. [example: Area-Minimizing Cones As Singular Models] In $\mathbb R^8=\mathbb R^4_x\times \mathbb R^4_y$, set \begin{align*} C=\{(x,y): |x|=|y|\}. \end{align*} The defining equation is \begin{align*} F(x,y)=|x|^2-|y|^2, \end{align*} so \begin{align*} \nabla F(x,y)=(2x,-2y). \end{align*} If $(x,y)\in C\setminus\{0\}$, then $|x|=|y|=\rho$ with $\rho>0$, and therefore \begin{align*} |\nabla F(x,y)|^2=|2x|^2+|-2y|^2=4|x|^2+4|y|^2=8\rho^2>0. \end{align*} Thus $C\setminus\{0\}$ is a smooth hypersurface of dimension $7$. On the smooth part, a unit normal is \begin{align*} \nu(x,y)=\frac{1}{\sqrt 2}\left(\frac{x}{|x|},-\frac{y}{|y|}\right). \end{align*} For $z\in \mathbb R^4\setminus\{0\}$, \begin{align*} \operatorname{div}\left(\frac{z}{|z|}\right)=\sum_{i=1}^4 \partial_{z_i}\left(\frac{z_i}{|z|}\right). \end{align*} Since \begin{align*} \partial_{z_i}\left(\frac{z_i}{|z|}\right)=\frac{1}{|z|}-\frac{z_i^2}{|z|^3}, \end{align*} summing over $i=1,\dots,4$ gives \begin{align*} \operatorname{div}\left(\frac{z}{|z|}\right)=\frac{4}{|z|}-\frac{|z|^2}{|z|^3}=\frac{3}{|z|}. \end{align*} Hence, on $C\setminus\{0\}$, \begin{align*} \operatorname{div}_{\mathbb R^8}\nu=\frac{1}{\sqrt 2}\left(\frac{3}{|x|}-\frac{3}{|y|}\right). \end{align*} Because $|x|=|y|$ on $C$, this becomes \begin{align*} \operatorname{div}_{\mathbb R^8}\nu=0. \end{align*} Thus the smooth part of $C$ has zero mean curvature. The nontrivial global fact, the *Bombieri-De Giorgi-Giusti theorem*, is that this minimal cone is area-minimizing as an integral $7$-current. The origin is singular because \begin{align*} \nabla F(0,0)=(0,0), \end{align*} and near $0$ the set is not a smooth embedded hypersurface. Every dilation fixes the cone: for $\lambda>0$, \begin{align*} (\lambda x,\lambda y)\in C \quad\Longleftrightarrow\quad |\lambda x|=|\lambda y| \quad\Longleftrightarrow\quad \lambda |x|=\lambda |y| \quad\Longleftrightarrow\quad |x|=|y|. \end{align*} Thus $C$ is a codimension-one area-minimizing $7$-current with an isolated interior singularity at the origin, so the codimension-one regularity theorem cannot be strengthened to rule out all singularities in dimension $m=7$. [/example] Cones appear because blow-up limits forget lower-order geometry and retain only scale-invariant behaviour. They are the local models against which every proposed regularity theorem must be tested. ## From Weak Minimizers To Classical Minimal Surfaces The final question is when the weak solution produced by currents satisfies the classical minimal surface equation. The answer is: at every regular point, and in low-dimensional cases the regular set may be the whole interior. [quotetheorem:5682] [citeproof:5682] This theorem is the rigorous version of the classical expectation for oriented soap films spanning a wire curve in $\mathbb R^3$: in dimension two and codimension one, the interior weak minimizer is a smooth minimal surface. The dimension and codimension hypotheses both carry weight. The two-dimensional codimension-one blow-up problem reduces to minimizing cones in $\mathbb R^3$, where the only interior minimizing cone is a plane; in dimension $7$ for hypersurfaces, the Simons cone gives a concrete model showing that cone classification cannot rule out singularities. The codimension-one restriction is also essential: in higher codimension, complex-analytic curves viewed as real integral currents can develop branch points, giving two-dimensional area-minimizing currents whose supports are not embedded smooth surfaces at the branch point. The theorem also says nothing about the boundary curve and does not select a unique minimizer. A smooth wire curve may still require separate boundary regularity arguments near $\operatorname{spt}\partial T$, and symmetric boundary data can admit more than one minimizing disk. Its role is to identify a low-dimensional setting where the singular-set estimates collapse to full interior smoothness. At a regular point in any dimension, the remaining task is to connect this smooth representative with the first variation formula from the start of the course. [quotetheorem:5683] [citeproof:5683] The weak and classical theories therefore agree wherever the regularity theory supplies a smooth representative. The regularity assumption is necessary in a concrete way: at the vertex of a minimizing cone, or at a branch point in higher codimension, there is no embedded smooth submanifold $M$ on which the classical mean curvature vector is defined. The interior assumption is also necessary because variations touching $\operatorname{spt}\partial T$ need not preserve the prescribed boundary. Area-minimizing is stronger than needed for this final implication; a smooth stationary current would already have $H_M=0$, but a smooth stable surface that is not stationary would not. In this chapter, minimization supplies both stationarity and the regularity input used before this theorem. The principle does not remove multiplicity and does not describe what happens on $\operatorname{sing} T$. It only translates the current into the classical Euler-Lagrange equation on the regular set. This limitation is also the forward connection: once regularity gives a graphical sheet, the current formulation hands the problem back to elliptic PDE. The next stage of the course uses the minimal surface system, Schauder and Sobolev estimates, and bootstrapping tools from earlier chapters to study curvature, compactness, and higher regularity on those smooth sheets. [example: Minimal Graph Recovered From A Current] Suppose an area-minimizing $m$-current is regular in a ball and, in local coordinates, its support is the graph of a smooth map $u:\Omega\subset \mathbb R^m\to\mathbb R^{n-m}$. Write the graph parametrization as \begin{align*} F(x)=(x,u(x)). \end{align*} For $i=1,\dots,m$, the tangent vector is \begin{align*} \partial_{x_i}F=e_i+\sum_{\alpha=1}^{n-m}\partial_{x_i}u_\alpha e_{m+\alpha}. \end{align*} Therefore the induced metric on the graph is \begin{align*} g_{ij}=\partial_{x_i}F\cdot \partial_{x_j}F=\left(e_i+\sum_{\alpha=1}^{n-m}\partial_{x_i}u_\alpha e_{m+\alpha}\right)\cdot\left(e_j+\sum_{\beta=1}^{n-m}\partial_{x_j}u_\beta e_{m+\beta}\right). \end{align*} Using $e_i\cdot e_j=\delta_{ij}$, $e_i\cdot e_{m+\beta}=0$, and $e_{m+\alpha}\cdot e_{m+\beta}=\delta_{\alpha\beta}$, this becomes \begin{align*} g_{ij}=\delta_{ij}+\sum_{\alpha=1}^{n-m}\partial_{x_i}u_\alpha\partial_{x_j}u_\alpha. \end{align*} The weak-to-classical minimal surface principle gives zero mean curvature for the smooth graph, so its area is stationary under compactly supported graph variations. Fix $\alpha\in\{1,\dots,n-m\}$ and choose $\varphi\in C_c^\infty(\Omega)$. For the variation $u_t=u+t\varphi e_\alpha$, the induced metric is \begin{align*} g_{ij}(t)=\delta_{ij}+\sum_{\beta=1}^{n-m}\partial_{x_i}(u_\beta+t\varphi\delta_{\alpha\beta})\partial_{x_j}(u_\beta+t\varphi\delta_{\alpha\beta}). \end{align*} Differentiating at $t=0$ gives \begin{align*} \frac{d}{dt}g_{ij}(t)\bigg|_{t=0}=\partial_{x_i}\varphi\,\partial_{x_j}u_\alpha+\partial_{x_i}u_\alpha\,\partial_{x_j}\varphi. \end{align*} Since $\frac{d}{dt}\sqrt{\det g(t)}\big|_{t=0}=\frac12\sqrt{\det g}\sum_{i,j=1}^m g^{ij}\frac{d}{dt}g_{ij}(t)\big|_{t=0}$, stationarity gives \begin{align*} 0=\frac12\int_\Omega \sqrt{\det g}\sum_{i,j=1}^m g^{ij}\left(\partial_{x_i}\varphi\,\partial_{x_j}u_\alpha+\partial_{x_i}u_\alpha\,\partial_{x_j}\varphi\right)\,dx. \end{align*} Because $g^{ij}=g^{ji}$, the two summands are equal after exchanging the indices $i$ and $j$, hence \begin{align*} 0=\int_\Omega \sqrt{\det g}\sum_{i,j=1}^m g^{ij}\partial_{x_j}u_\alpha\,\partial_{x_i}\varphi\,dx. \end{align*} Integrating by parts produces no boundary term because $\varphi$ is compactly supported, so \begin{align*} 0=-\int_\Omega \varphi\sum_{i=1}^m\partial_{x_i}\left(\sqrt{\det g}\sum_{j=1}^m g^{ij}\partial_{x_j}u_\alpha\right)\,dx. \end{align*} Since this holds for every $\varphi\in C_c^\infty(\Omega)$, the fundamental lemma gives \begin{align*} \sum_{i=1}^m\partial_{x_i}\left(\sqrt{\det g}\sum_{j=1}^m g^{ij}\partial_{x_j}u_\alpha\right)=0,\qquad \alpha=1,\dots,n-m. \end{align*} Thus the regular current is governed locally by the minimal surface system for its graphing function $u$, so the weak current solution has returned to the classical elliptic PDE description. [/example] The chapter's logic is now complete. Currents solve the spanning problem by the direct method, Allard's theorem detects smooth graphical regions, dimension-reduction controls the remaining singular set, and low-dimensional area minimizers recover the classical minimal surfaces that motivated Plateau's problem. The Plateau problem closes the first half of the course by showing how direct methods, currents, and regularity theory produce genuine area-minimizing surfaces. The second half now shifts to the parallel variational theory of maps, where the same ideas reappear for the Dirichlet energy instead of area. # 7. Harmonic Maps And The Tension Field The second half of the course changes the variational object introduced in Chapter 0: instead of varying submanifolds and measuring area, we vary maps and measure how much they stretch tangent vectors. The Dirichlet energy is the analytic functional behind harmonic functions, geodesics, and harmonic maps between curved spaces. This chapter sets up the energy for Sobolev maps between Riemannian manifolds, computes its first variation, and rewrites the critical point equation as a nonlinear elliptic system in local coordinates. ## Measuring Energy For Maps Between Manifolds The basic question is how to assign a first-derivative energy to a map $u:M\to N$ when both domain and target are curved. For maps into Euclidean space, the Dirichlet energy integrates $|\nabla u|^2$ over the domain. For a manifold-valued map, the derivative $du_x:T_xM\to T_{u(x)}N$ still has a Hilbert-Schmidt norm once the metrics on $M$ and $N$ are fixed, and the energy is the integral of this norm. [definition: Energy Density Of A Smooth Map] Let $(M,g)$ and $(N,h)$ be Riemannian manifolds and let $u:M\to N$ be a smooth map. The energy density of $u$ is the function $e(u):M\to \mathbb R$ defined by \begin{align*} e(u)(x)=\frac{1}{2}|du_x|_{g,h}^2, \end{align*} where $|du_x|_{g,h}^2$ is the Hilbert-Schmidt norm of the linear map $du_x:T_xM\to T_{u(x)}N$ with respect to $g_x$ and $h_{u(x)}$. [/definition] The factor $1/2$ is not mathematically essential, but it removes a factor of $2$ from the first variation. In a local $g$-orthonormal frame $(e_1,\dots,e_m)$ on $M$, the same density is \begin{align*} e(u)=\frac{1}{2}\sum_{i=1}^m h(du(e_i),du(e_i)). \end{align*} This formula shows that the definition is independent of the chosen orthonormal frame. [definition: Dirichlet Energy Of A Smooth Map] Let $(M,g)$ be a Riemannian manifold with Riemannian volume measure $d\operatorname{vol}_g$, and let $(N,h)$ be a Riemannian manifold. The Dirichlet energy is the functional \begin{align*} E:C^\infty(M,N)\to [0,\infty] \end{align*} defined by \begin{align*} E[u]=\int_M e(u)\,d\operatorname{vol}_g =\frac{1}{2}\int_M |du|_{g,h}^2\,d\operatorname{vol}_g. \end{align*} [/definition] For variational arguments, smooth maps are too restrictive: minimising sequences often converge only weakly. The Sobolev class $W^{1,2}(M,N)$ is the natural finite-energy space, and its definition is usually made by embedding $N$ isometrically into some Euclidean space and imposing the pointwise constraint almost everywhere. [definition: Sobolev Map Into A Riemannian Manifold] Let $(M,g)$ be a Riemannian manifold, let $(N,h)$ be a compact Riemannian manifold, and fix an isometric embedding $\iota:N\hookrightarrow \mathbb R^K$. A measurable map $u:M\to N$ belongs to $W^{1,2}(M,N)$ if $\iota\circ u\in W^{1,2}(M;\mathbb R^K)$ and $\iota(u(x))\in \iota(N)$ for a.e. $x\in M$. [/definition] The embedding is a convenient way to use standard Sobolev spaces, but the resulting class and energy are intrinsic. The differential of a Sobolev map is interpreted weakly after embedding, and its tangential component along $N$ gives the geometric [weak derivative](/page/Weak%20Derivative). [example: Finite Energy Circle-Valued Map] Let $M=S^1$ with arclength coordinate $t\in[0,2\pi]$ and let $N=S^1\subset\mathbb R^2$. For an integer $k$, define \begin{align*} u(t)=e^{ikt}=(\cos(kt),\sin(kt)). \end{align*} The identity $\cos^2(kt)+\sin^2(kt)=1$ gives $u(t)\in S^1$ for every $t$, and the coordinate functions are smooth, so $u\in W^{1,2}(S^1,S^1)$. Differentiating componentwise, \begin{align*} u'(t)=(-k\sin(kt),k\cos(kt)). \end{align*} Since $S^1\subset\mathbb R^2$ has the induced Euclidean metric, its speed squared is \begin{align*} |u'(t)|^2=(-k\sin(kt))^2+(k\cos(kt))^2. \end{align*} Expanding the squares and using $\sin^2(kt)+\cos^2(kt)=1$ gives \begin{align*} |u'(t)|^2=k^2\sin^2(kt)+k^2\cos^2(kt)=k^2. \end{align*} Therefore the Dirichlet energy is \begin{align*} E[u]=\frac{1}{2}\int_0^{2\pi}|u'(t)|^2\,dt. \end{align*} Substituting $|u'(t)|^2=k^2$, \begin{align*} E[u]=\frac{1}{2}\int_0^{2\pi}k^2\,dt. \end{align*} Since $k^2$ is constant in $t$, \begin{align*} E[u]=\frac{1}{2}(2\pi k^2)=\pi k^2. \end{align*} This example records how the energy detects winding: in this symmetric representative, increasing $|k|$ forces the derivative length to be larger at every point. [/example] The same energy that measures winding on a circle also generalises the classical Dirichlet integral for scalar functions. This connection is the first consistency check for the definition. [example: Harmonic Functions As Real-Valued Harmonic Maps] Let $(M,g)$ be a compact Riemannian manifold and take $N=\mathbb R$ with its Euclidean metric $dy^2$. For a smooth function $u:M\to\mathbb R$, choose a local $g$-orthonormal frame $(e_1,\dots,e_m)$. Since \begin{align*} du(e_i)=e_i(u)\,\partial_y \end{align*} and $dy^2(\partial_y,\partial_y)=1$, each summand in the energy density is \begin{align*} dy^2\bigl(du(e_i),du(e_i)\bigr)=dy^2\bigl(e_i(u)\partial_y,e_i(u)\partial_y\bigr)=e_i(u)^2. \end{align*} Therefore \begin{align*} e(u)=\frac{1}{2}\sum_{i=1}^m e_i(u)^2. \end{align*} By the defining identity $g(\nabla u,X)=X(u)$ for the Riemannian gradient, we have $g(\nabla u,e_i)=e_i(u)$ for each $i$, so \begin{align*} |\nabla u|_g^2=\sum_{i=1}^m g(\nabla u,e_i)^2=\sum_{i=1}^m e_i(u)^2. \end{align*} Substituting this into the definition of the Dirichlet energy gives \begin{align*} E[u]=\int_M e(u)\,d\operatorname{vol}_g=\frac{1}{2}\int_M |\nabla u|_g^2\,d\operatorname{vol}_g. \end{align*} Thus the harmonic map energy reduces exactly to the usual Dirichlet energy for real-valued functions. Since the Euclidean metric on $\mathbb R$ has zero Christoffel symbols in the coordinate $y$, the later [coordinate formula for the tension field](/theorems/5686) becomes \begin{align*} \tau(u)=\Delta_g u. \end{align*} Hence the harmonic map equation $\tau(u)=0$ is precisely the classical harmonic function equation $\Delta_g u=0$. [/example] ## First Variation And The Tension Field The next problem is to identify the Euler-Lagrange equation for $E$. Since the target is curved, a variation of $u$ is not an arbitrary vector-valued perturbation; its velocity is a section of the pulled-back tangent bundle $u^{-1}TN$. The first variation pairs this variation field against a canonical section called the tension field. [definition: Variation Field Of A Map] Let $u:M\to N$ be a smooth map. A smooth variation of $u$ is a smooth map $U:(-\varepsilon,\varepsilon)\times M\to N$ such that $U(0,x)=u(x)$ for all $x\in M$. Its variation field is the section $V\in\Gamma(u^{-1}TN)$ defined by \begin{align*} V(x)=\frac{\partial U}{\partial s}(0,x)\in T_{u(x)}N. \end{align*} [/definition] The variation field is the geometric analogue of a test function in the scalar calculus of variations. To express the first variation as a pairing against $V$ itself rather than against its derivative, we need the geometric trace of the second covariant derivative of $u$. [definition: Tension Field] Let $u:(M,g)\to (N,h)$ be smooth. The tension field of $u$ is the section $\tau(u)\in\Gamma(u^{-1}TN)$ defined by \begin{align*} \tau(u)=\operatorname{tr}_g \nabla du, \end{align*} where $\nabla du$ is the covariant derivative of the differential $du\in\Gamma(T^*M\otimes u^{-1}TN)$ using the Levi-Civita connections of $M$ and $N$. [/definition] Equivalently, the tension construction assigns to each smooth map its own section of the corresponding pullback tangent bundle: \begin{align*} \tau(u)\in \Gamma(u^{-1}TN), \qquad \tau(u)=\operatorname{tr}_g\nabla du. \end{align*} Thus $\tau$ is best viewed as a nonlinear geometric operator over the mapping space, rather than as a map into one fixed vector space. This definition is compact, but it is worth unpacking the trace before varying the energy. If $(e_1,\dots,e_m)$ is a local $g$-orthonormal frame, then \begin{align*} \tau(u)=\sum_{i=1}^m \left(\nabla^N_{du(e_i)}du(e_i)-du(\nabla^M_{e_i}e_i)\right). \end{align*} The expression is independent of the chosen frame because it is the metric trace of a tensorial object. [quotetheorem:5684] [citeproof:5684] The theorem identifies $\tau(u)$ as the $L^2$-gradient of the energy, up to sign, but only under the closed-domain hypotheses stated above. Compactness ensures that differentiating under the integral and integrating by parts do not introduce decay conditions at infinity, while the absence of boundary removes the boundary term that would otherwise contain the normal derivative of the variation. A concrete boundary failure already appears for scalar maps on $M=[0,1]$ with $N=\mathbb R$: the same calculation gives \begin{align*} \frac{d}{ds}\Big|_{s=0}E[u+sV] =-\int_0^1 u''V\,dt+u'(1)V(1)-u'(0)V(0), \end{align*} so the displayed closed-manifold formula is false unless the boundary values are fixed or the endpoint terms are imposed as natural boundary conditions. A noncompact failure appears on $M=\mathbb R$ for $u(t)=t$ and compactly supported variations: the local integration by parts is meaningful, but the energy $E[u]=\infty$, so the functional in the theorem is not a finite differentiable functional on that map. The theorem is therefore a first-variation formula, not an existence theorem and not a regularity theorem. It motivates naming the critical maps: the analogue of a minimal submanifold is a map whose energy-gradient section vanishes. [definition: Harmonic Map] Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. A smooth map $u:M\to N$ is a harmonic map if \begin{align*} \tau(u)=0. \end{align*} [/definition] For weak maps, the same equation is imposed by testing the first variation against compactly supported variation fields. To justify the definition in the smooth setting, we now prove that it is equivalent to the original variational condition. [quotetheorem:5685] [citeproof:5685] This result is the exact analogue of minimality implying zero mean curvature in the first half of the course: the mean curvature vector is the gradient of area for immersions, while the tension field is the gradient of Dirichlet energy for maps. The smoothness hypothesis matters because the proof uses pointwise variation fields and the exponential map to realise arbitrary local sections of $u^{-1}TN$ as velocities of variations. If regularity is removed, pointwise vanishing of $\tau(u)$ may not even be a defined statement: for the scalar map $u(x)=|x|$ on $(-1,1)$, the weak derivative lies in $L^2$, but the distributional second derivative is $2\delta_0$, so the classical equation $\Delta u=0$ fails to be a pointwise equation. On a domain with boundary, the equivalence also changes with the admissible variations. For $u(t)=t$ on $[0,1]$ as a map to $\mathbb R$, one has $u''=0$, but the first variation under free endpoint variations is $V(1)-V(0)$, so $u$ is not critical unless endpoint variations are restricted or Neumann conditions are added. Finally, criticality is weaker than minimisation: a harmonic map may be unstable or have larger energy than another map in the same homotopy class. [example: Geodesics As Harmonic Maps From Intervals] Let $I\subset\mathbb R$ have coordinate $t$ and Euclidean metric $dt^2$, and let $\gamma:I\to N$ be smooth. The vector field $\partial_t$ is a unit frame on $I$, and the Euclidean Levi-Civita connection satisfies $\nabla^I_{\partial_t}\partial_t=0$. Since \begin{align*} d\gamma(\partial_t)=\dot\gamma, \end{align*} the frame formula for the tension field gives \begin{align*} \tau(\gamma)=\nabla^N_{d\gamma(\partial_t)}d\gamma(\partial_t)-d\gamma(\nabla^I_{\partial_t}\partial_t). \end{align*} Substituting the two identities above, \begin{align*} \tau(\gamma)=\nabla^N_{\dot\gamma}\dot\gamma-d\gamma(0). \end{align*} Because $d\gamma$ is linear on each tangent space, $d\gamma(0)=0$, so \begin{align*} \tau(\gamma)=\nabla^N_{\dot\gamma}\dot\gamma. \end{align*} Therefore \begin{align*} \tau(\gamma)=0 \Longleftrightarrow \nabla^N_{\dot\gamma}\dot\gamma=0. \end{align*} Thus a curve is harmonic as a map from the interval exactly when it satisfies the geodesic equation. The energy density has only one unit direction to trace over, so \begin{align*} e(\gamma)(t)=\frac{1}{2}h(\dot\gamma(t),\dot\gamma(t)). \end{align*} Hence \begin{align*} E[\gamma]=\frac{1}{2}\int_I h(\dot\gamma,\dot\gamma)\,dt, \end{align*} which is the usual energy of a parametrised curve. If $\gamma$ is harmonic, metric compatibility of the Levi-Civita connection gives \begin{align*} \frac{d}{dt}h(\dot\gamma,\dot\gamma)=h(\nabla^N_{\dot\gamma}\dot\gamma,\dot\gamma)+h(\dot\gamma,\nabla^N_{\dot\gamma}\dot\gamma). \end{align*} By symmetry of $h$, \begin{align*} \frac{d}{dt}h(\dot\gamma,\dot\gamma)=2h(\nabla^N_{\dot\gamma}\dot\gamma,\dot\gamma). \end{align*} For a harmonic curve, $\nabla^N_{\dot\gamma}\dot\gamma=0$, so \begin{align*} \frac{d}{dt}|\dot\gamma|_h^2=0. \end{align*} Thus the harmonic map equation recovers geodesics with constant speed. [/example] ## The Local Coordinate System The geometric equation $\tau(u)=0$ is concise, but elliptic PDE estimates require a coordinate formula. The issue is that the highest-order part should look like a Laplace-Beltrami operator, while the target curvature enters through lower-order quadratic terms in the first derivatives of $u$. Let $(x_1,\dots,x_m)$ be local coordinates on $M$ and $(y_1,\dots,y_n)$ local coordinates on $N$. Let $u$ be smooth on the coordinate patch and write its coordinate expression as $u=(u_1,\dots,u_n)$. Let $g_{ij}$ be the metric coefficients on $M$, let $g^{ij}$ be the inverse matrix, and let $\Gamma^\alpha_{\beta\gamma}$ be the Christoffel symbols of $N$ in the chosen target chart. [quotetheorem:5686] [citeproof:5686] This theorem turns the harmonic map equation into a semilinear elliptic system: \begin{align*} \Delta_g u_\alpha +g^{ij}\Gamma^\alpha_{\beta\gamma}(u)\,\partial_{x_i}u_\beta\,\partial_{x_j}u_\gamma=0, \qquad \alpha=1,\dots,n. \end{align*} The leading term is elliptic because $g^{ij}$ is positive definite, while the nonlinearity is quadratic in $du$. The formula is local in two senses: the domain coordinates must remain inside a coordinate patch on $M$, and the image of $u$ must lie inside the chosen target chart for the component functions $u_\alpha$ to be valid. The target-chart hypothesis has a concrete obstruction: the identity map $S^1\to S^1$ cannot be represented by a single angle function on all of $S^1$, so no global single-coordinate formula of the displayed form exists even though the intrinsic tension field is globally defined. The smoothness hypothesis is also needed for the pointwise expression; for $u(x)=|x|$ as a map $(-1,1)\to\mathbb R$, the second derivative appearing in $\Delta u$ is a distribution rather than a function. Positive definiteness of $g$ is essential for ellipticity: replacing the domain metric by the Lorentzian form $dt^2-dx^2$ changes the leading operator to the wave operator $\partial_t^2-\partial_x^2$, and the elliptic estimates used for harmonic maps no longer apply. The components $u_\alpha$ and Christoffel symbols are coordinate-dependent, although the section $\tau(u)$ is intrinsic. Ellipticity of the leading part alone does not give smoothness for weak solutions, since the quadratic term may have low integrability at the natural energy level. [example: Harmonic Maps Into Euclidean Space] Let $N=\mathbb R^n$ with its Euclidean metric in the standard coordinates $(y_1,\dots,y_n)$, and write a smooth map $u:M\to\mathbb R^n$ as \begin{align*} u=(u_1,\dots,u_n). \end{align*} For the Euclidean metric, the coordinate vector fields $\partial_{y_1},\dots,\partial_{y_n}$ are constant, so the Euclidean Levi-Civita connection satisfies \begin{align*} \nabla^{\mathbb R^n}_{\partial_{y_\beta}}\partial_{y_\gamma}=0. \end{align*} The Christoffel symbols are defined by \begin{align*} \nabla^{\mathbb R^n}_{\partial_{y_\beta}}\partial_{y_\gamma}=\Gamma^\alpha_{\beta\gamma}\partial_{y_\alpha}. \end{align*} Comparing the last two identities gives \begin{align*} \Gamma^\alpha_{\beta\gamma}=0 \end{align*} for every $\alpha,\beta,\gamma$ in these coordinates. By the *Coordinate Formula For The Tension Field*, \begin{align*} \tau(u)_\alpha=\Delta_g u_\alpha+g^{ij}\Gamma^\alpha_{\beta\gamma}(u)\,\partial_{x_i}u_\beta\,\partial_{x_j}u_\gamma. \end{align*} Substituting $\Gamma^\alpha_{\beta\gamma}=0$ gives \begin{align*} \tau(u)_\alpha=\Delta_g u_\alpha+g^{ij}\cdot 0\cdot \partial_{x_i}u_\beta\,\partial_{x_j}u_\gamma. \end{align*} Since multiplication by $0$ annihilates the quadratic term, \begin{align*} \tau(u)_\alpha=\Delta_g u_\alpha. \end{align*} Therefore $\tau(u)=0$ holds exactly when every component satisfies \begin{align*} \Delta_g u_\alpha=0 \end{align*} for $\alpha=1,\dots,n$. Thus a map into Euclidean space is harmonic exactly when each of its coordinate functions is a harmonic function on $(M,g)$. [/example] The Euclidean example hides the main geometric feature: for a curved target, the components of $u$ interact through the Christoffel symbols. Even though the equation is second-order elliptic, it is a system rather than a scalar equation, and regularity depends on controlling the quadratic first-derivative term. [example: Identity Maps Between Conformal Surfaces] Let $(\Sigma,g)$ be an oriented surface and let $\tilde g=e^{2\varphi}g$. For the identity map $u=\operatorname{id}:(\Sigma,g)\to(\Sigma,\tilde g)$, choose local coordinates $(x_1,x_2)$ on the domain and use the same coordinate functions on the target. Then, for $\alpha,\beta\in\{1,2\}$, \begin{align*} u_\alpha(x)=x_\alpha,\qquad \partial_{x_i}u_\beta=\delta_{i\beta}. \end{align*} The coordinate formula for the tension field, applied with the target Christoffel symbols $\tilde\Gamma^\alpha_{\beta\gamma}$ of $\tilde g$, gives \begin{align*} \tau(u)_\alpha=\Delta_g u_\alpha+g^{ij}\tilde\Gamma^\alpha_{\beta\gamma}(u)\,\partial_{x_i}u_\beta\,\partial_{x_j}u_\gamma. \end{align*} Substituting $u_\alpha=x_\alpha$ and $\partial_{x_i}u_\beta=\delta_{i\beta}$ gives \begin{align*} \tau(u)_\alpha=\Delta_g x_\alpha+g^{ij}\tilde\Gamma^\alpha_{\beta\gamma}\delta_{i\beta}\delta_{j\gamma}. \end{align*} The Kronecker deltas force $\beta=i$ and $\gamma=j$, so \begin{align*} \tau(u)_\alpha=\Delta_g x_\alpha+g^{ij}\tilde\Gamma^\alpha_{ij}. \end{align*} For the domain metric $g$, the Laplace-Beltrami operator on a coordinate function satisfies \begin{align*} \Delta_g x_\alpha=-g^{ij}\Gamma^\alpha_{ij}, \end{align*} where $\Gamma^\alpha_{ij}$ are the Christoffel symbols of $g$. Hence \begin{align*} \tau(u)_\alpha=g^{ij}\left(\tilde\Gamma^\alpha_{ij}-\Gamma^\alpha_{ij}\right). \end{align*} For the conformal change $\tilde g=e^{2\varphi}g$, the Christoffel symbols are related by \begin{align*} \tilde\Gamma^\alpha_{ij}=\Gamma^\alpha_{ij}+\delta^\alpha_i\partial_{x_j}\varphi+\delta^\alpha_j\partial_{x_i}\varphi-g_{ij}g^{\alpha\ell}\partial_{x_\ell}\varphi. \end{align*} Therefore \begin{align*} \tilde\Gamma^\alpha_{ij}-\Gamma^\alpha_{ij}=\delta^\alpha_i\partial_{x_j}\varphi+\delta^\alpha_j\partial_{x_i}\varphi-g_{ij}g^{\alpha\ell}\partial_{x_\ell}\varphi. \end{align*} Substituting this into the expression for $\tau(u)_\alpha$ gives \begin{align*} \tau(u)_\alpha=g^{ij}\delta^\alpha_i\partial_{x_j}\varphi+g^{ij}\delta^\alpha_j\partial_{x_i}\varphi-g^{ij}g_{ij}g^{\alpha\ell}\partial_{x_\ell}\varphi. \end{align*} The first contraction is \begin{align*} g^{ij}\delta^\alpha_i\partial_{x_j}\varphi=g^{\alpha j}\partial_{x_j}\varphi. \end{align*} The second contraction is \begin{align*} g^{ij}\delta^\alpha_j\partial_{x_i}\varphi=g^{i\alpha}\partial_{x_i}\varphi. \end{align*} Since $g^{ij}=g^{ji}$, renaming the dummy index in the second term gives \begin{align*} g^{i\alpha}\partial_{x_i}\varphi=g^{\alpha\ell}\partial_{x_\ell}\varphi. \end{align*} On a surface, $g^{ij}g_{ij}=2$, because this is the trace of the identity endomorphism of a two-dimensional tangent space. Thus \begin{align*} \tau(u)_\alpha=g^{\alpha\ell}\partial_{x_\ell}\varphi+g^{\alpha\ell}\partial_{x_\ell}\varphi-2g^{\alpha\ell}\partial_{x_\ell}\varphi. \end{align*} The three terms cancel, so \begin{align*} \tau(u)_\alpha=0. \end{align*} This holds for each $\alpha=1,2$, hence \begin{align*} \tau(u)=0. \end{align*} Thus $\operatorname{id}:(\Sigma,g)\to(\Sigma,e^{2\varphi}g)$ is harmonic. The cancellation is exactly two-dimensional: the two first-derivative terms coming from the conformal factor are cancelled by the trace term $g^{ij}g_{ij}=2$. [/example] ## Weak Formulation And Elliptic Character The final issue in this chapter is how the smooth equation interacts with Sobolev compactness. Variational limits naturally live in $W^{1,2}(M,N)$, so the Euler-Lagrange equation should be expressible without assuming second derivatives at the start. Pointwise smooth convergence is usually unavailable for minimizing sequences: weak $W^{1,2}$ compactness controls the first derivative in an integral norm, not [uniform convergence](/page/Uniform%20Convergence) of derivatives. A naive componentwise Sobolev definition also depends on coordinates, because changing target charts applies a nonlinear transition map to the components; the embedding definition packages the constraint in a coordinate-independent way. [definition: Weakly Harmonic Map] Let $(M,g)$ be a Riemannian manifold, let $(N,h)$ be a compact Riemannian manifold isometrically embedded in $\mathbb R^K$, and let $u\in W^{1,2}(M,N)$. The map $u$ is weakly harmonic if the first variation of $E$ vanishes for every compactly supported smooth variation through maps into $N$. [/definition] This definition is intrinsic, but the embedding viewpoint gives a useful analytic form. If $A$ is the second fundamental form of $N\subset\mathbb R^K$, the weak equation can be written schematically as a Euclidean elliptic system with a normal quadratic term. [quotetheorem:5687] [citeproof:5687] This weak equation is the starting point for the later regularity chapters. Compactness of $N$ and the embedding hypothesis keep the constraint set geometrically controlled and allow the second fundamental form to be treated as a bounded smooth tensor along $u$. In estimates, the equation is used by testing against cut-off variations, projecting Euclidean test fields onto $T_uN$, and applying elliptic estimates to $\Delta_g u$ while treating $A(u)(\nabla u,\nabla u)$ as the nonlinear error term. In dimension two the quadratic structure is borderline: it interacts with compensation phenomena and conservation-law rewritings, rather than behaving like an ordinary higher-integrability source term. These mechanisms connect harmonic map theory to the regularity theory of elliptic systems and to the geometric analysis theme already seen for minimal submanifolds, where a variational equation is converted into analytic estimates for curvature or derivative quantities. If the compactness or bounded-geometry control is dropped, the quadratic coefficient can become uncontrolled: for a noncompact embedded target with unbounded second fundamental form, a finite-energy map whose image enters high-curvature regions may make $A(u)(\nabla u,\nabla u)$ fail to have the expected local integrability. The manifold constraint is also essential; if $u$ is merely an unconstrained $\mathbb R^K$-valued Sobolev map, there is no normal bundle or second fundamental form term, and the weak critical equation is instead the linear equation $\Delta_g u=0$. The displayed sign depends on the convention for $A$; authors using the opposite convention write the same equation with the opposite sign on the quadratic term. The theorem does not assert that a weakly harmonic map is smooth, nor does it remove the manifold constraint from the problem. Its leading operator is linear elliptic, but the term $A(u)(\nabla u,\nabla u)$ lies only in $L^1$ when $u\in W^{1,2}$, which is exactly the borderline difficulty in two-dimensional harmonic map regularity. Harmonic maps generalize the variational viewpoint from submanifolds to maps, replacing area by energy and mean curvature by the tension field. The next chapter asks how curvature of the target and source influences the resulting elliptic equation through Bochner identities and comparison principles. # 8. Bochner Identities And Curvature Conditions This chapter brings together the analytic and geometric sides of harmonic map theory. In the preceding chapter, harmonic maps appeared as critical points of the Dirichlet energy and hence as solutions of the tension field equation $\tau(u)=0$. We now ask what extra information comes from applying a Weitzenbock-type identity to $|du|^2$: curvature of the domain and target enters the second-order behaviour of the energy density, and that sign information drives subharmonicity, uniqueness, and rigidity. The guiding principle is that nonpositive sectional curvature in the target prevents the differential of a harmonic map from concentrating in a convex direction, while positive Ricci curvature on the domain pushes harmonic maps toward constancy. This is the harmonic-map analogue of Chapter 2's [stability inequality for minimal hypersurfaces](/theorems/5659), where second variation controlled curvature through a coercive estimate. ## The Bochner Formula For Harmonic Maps The basic problem is to convert the nonlinear harmonic map equation into a scalar differential inequality. Since the equation $\tau(u)=0$ takes values in the pullback bundle $u^{-1}TN$, it is not directly ordered. The energy density is scalar, so applying the Laplacian to it creates a place where maximum principles can be used. Let $(M,g)$ and $(N,h)$ be smooth Riemannian manifolds, and let $u:M\to N$ be smooth. The first scalar quantity attached to $u$ is the pointwise norm of its differential. [definition: Energy Density Of A Smooth Map] Let $u:(M,g)\to (N,h)$ be a smooth map. The energy density of $u$ is the function $e(u):M\to\mathbb R$ defined by \begin{align*} e(u)(p) = \frac{1}{2}|du_p|^2, \end{align*} where $|du_p|^2$ is computed using $g$ on $T_pM$ and $h$ on $T_{u(p)}N$. [/definition] Thus if $e_1,\dots,e_m$ is a local $g$-orthonormal frame on $M$, then \begin{align*} e(u)=\frac{1}{2}\sum_{i=1}^m |du(e_i)|_h^2. \end{align*} The factor $1/2$ is chosen so that integrating $e(u)$ gives the Dirichlet energy used in the Euler-Lagrange equation. [example: Energy Density Of Maps Into A Flat Torus] Let $N=\mathbb R^k/\Lambda$ be a flat torus, let $u:M\to N$ be smooth, and choose an open set $U\subset M$ on which $u$ lifts to $\tilde u=(\tilde u_1,\dots,\tilde u_k):U\to\mathbb R^k$. If $e_1,\dots,e_m$ is a local $g$-orthonormal frame on $U$, then the quotient map $\pi:\mathbb R^k\to\mathbb R^k/\Lambda$ is a local isometry, so for each $i$, \begin{align*} |du(e_i)|_h^2=|d\tilde u(e_i)|_{\mathbb R^k}^2=\left|\sum_{\alpha=1}^k e_i(\tilde u_\alpha)\partial_\alpha\right|_{\mathbb R^k}^2=\sum_{\alpha=1}^k e_i(\tilde u_\alpha)^2. \end{align*} Using the definition of energy density and then interchanging the finite sums gives \begin{align*} e(u)=\frac{1}{2}\sum_{i=1}^m |du(e_i)|_h^2=\frac{1}{2}\sum_{i=1}^m\sum_{\alpha=1}^k e_i(\tilde u_\alpha)^2=\frac{1}{2}\sum_{\alpha=1}^k\sum_{i=1}^m e_i(\tilde u_\alpha)^2. \end{align*} Since $\sum_{i=1}^m e_i(\tilde u_\alpha)^2=|d\tilde u_\alpha|^2$, this becomes \begin{align*} e(u)=\frac{1}{2}\sum_{\alpha=1}^k |d\tilde u_\alpha|^2. \end{align*} In the same lifted coordinates, the Euclidean connection satisfies $\nabla^{\mathbb R^k}_{\partial_\alpha}\partial_\beta=0$. Therefore, for each frame vector $e_i$, \begin{align*} \nabla^{\mathbb R^k}_{e_i}(d\tilde u(e_i))=\sum_{\alpha=1}^k e_i(e_i(\tilde u_\alpha))\partial_\alpha. \end{align*} Also, \begin{align*} d\tilde u(\nabla^M_{e_i}e_i)=\sum_{\alpha=1}^k (\nabla^M_{e_i}e_i)(\tilde u_\alpha)\partial_\alpha. \end{align*} Taking the trace in the definition of the tension field gives \begin{align*} \tau(\tilde u)=\sum_{\alpha=1}^k\left(\sum_{i=1}^m e_i(e_i(\tilde u_\alpha))-(\nabla^M_{e_i}e_i)(\tilde u_\alpha)\right)\partial_\alpha=\sum_{\alpha=1}^k(\Delta_g\tilde u_\alpha)\partial_\alpha. \end{align*} Thus $u$ is harmonic on $U$ exactly when each lifted component satisfies $\Delta_g\tilde u_\alpha=0$. The local calculation is Euclidean, while the global topology of the map is recorded by how the local lifts change by elements of the lattice $\Lambda$. [/example] The flat torus example also shows the limitation of componentwise notation: for a curved target, the differential takes values in a moving tangent space. To compute $\Delta_g e(u)$ invariantly, we need a connection on the pullback bundle and a covariant derivative of $du$. [definition: Covariant Derivative Of The Differential] Let $u:(M,g)\to(N,h)$ be smooth. The covariant derivative $\nabla du$ is the section of $T^*M\otimes T^*M\otimes u^{-1}TN$ defined by \begin{align*} (\nabla du)(X,Y)=\nabla^N_X(du(Y))-du(\nabla^M_XY), \end{align*} for vector fields $X,Y$ on $M$. [/definition] The trace of this tensor is the tension field. Harmonicity removes that trace from the Bochner identity, but the full tensor $\nabla du$ remains as a nonnegative square term, so the next theorem is the identity that makes curvature visible in the scalar function $e(u)$. [quotetheorem:5688] [citeproof:5688] The formula separates the analytic positivity $|\nabla du|^2$ from the two geometric signs. Domain Ricci curvature appears with a favourable sign when $\operatorname{Ric}^M\ge 0$, while target sectional curvature appears with a favourable sign when $K_N\le 0$; the following sign check is what allows the theorem to become an inequality. These hypotheses are not decorative: if the target has positive sectional curvature, the target-curvature term can enter with the opposite sign, and the energy density need not be subharmonic. If the domain has negative Ricci curvature, the Ricci contribution can also defeat the square term. The Bochner formula therefore does not prove regularity or rigidity by itself; it becomes useful only after the curvature signs convert the identity into a maximum-principle inequality. [remark: Sign Of The Target Curvature Term] For orthonormal vectors $v,w\in T_qN$, the sectional curvature convention gives \begin{align*} h(R^N(v,w)w,v)=K_N(v\wedge w). \end{align*} Thus if $K_N\le 0$, the final term in the Bochner formula is nonnegative after the preceding minus sign. This is the source of the subharmonicity estimates in the next section. [/remark] ## Subharmonicity Under Nonpositive Target Curvature The next question is when the Bochner identity becomes an inequality of the form $\Delta_g e(u)\ge 0$. Such an inequality turns the energy density into a subharmonic function, so local mean-value estimates and global maximum principles become available. [definition: Nonpositive Sectional Curvature] A Riemannian manifold $(N,h)$ has nonpositive sectional curvature if for every $q\in N$ and every $2$-plane $\sigma\subset T_qN$, \begin{align*} K_N(\sigma)\le 0. \end{align*} [/definition] This condition is exactly what is needed to control the target term in the Bochner formula. If the domain Ricci curvature is also nonnegative, then all curvature terms have the same sign, and the scalar maximum principle becomes available for harmonic maps. [quotetheorem:5689] [citeproof:5689] This is the central point where curvature becomes an analytic maximum-principle hypothesis. The assumptions are close to sharp for this method: if $M$ has negative Ricci curvature, the Ricci term can be negative, and if $N$ has positive sectional curvature, the target term can also be negative after the sign in the Bochner formula is applied. Maps from flat tori to flat tori show another limitation: even when all curvature terms vanish and $e(u)$ is subharmonic, nonconstant harmonic maps can remain because parallel differentials are allowed. The next step is to see what a subharmonic energy density can do on a compact boundaryless domain, where integration and the maximum principle leave no room for a positive Laplacian. [quotetheorem:5690] [citeproof:5690] The conclusion is rigidity for the differential rather than immediate rigidity for the map. A parallel differential has constant length and is transported consistently by the Levi-Civita connections, so the map behaves like an affine map along geodesic directions rather than like an arbitrary harmonic map. This is already strong enough to rule out many sources of oscillation, but it still allows nonconstant examples when the curvature terms in the Bochner identity vanish identically, such as maps between flat tori. The remaining obstruction is therefore the possible survival of a nonzero parallel differential. To obtain constancy, we ask what happens when the domain has a point where Ricci curvature is strictly positive: at such a point, any nonzero differential would contribute a strictly positive Ricci term, contradicting the vanishing forced by the integrated Bochner argument. [quotetheorem:5691] [citeproof:5691] This theorem is the cleanest rigidity consequence of the Bochner method. The standard sphere gives the model case, and it also makes visible why the conclusion depends on the domain curvature rather than on compactness of the target. [example: Constant Maps From Positively Ricci Curved Domains] Let $m\ge 2$, let $M=S^m$ have its round metric of sectional curvature $1$, and let $N$ be any complete manifold with $K_N\le 0$. For the round sphere, \begin{align*} \operatorname{Ric}^{S^m}=(m-1)g. \end{align*} Thus, if $X\in T_pS^m$ and $X\neq 0$, then \begin{align*} \operatorname{Ric}^{S^m}(X,X)=(m-1)g(X,X)=(m-1)|X|^2>0, \end{align*} so the Ricci curvature is positive definite at every point. Let $u:S^m\to N$ be smooth and harmonic. By *Energy Density Constancy On Compact Domains*, $e(u)$ is constant and $\nabla du=0$, so $\Delta_g e(u)=0$. Substituting these facts into the *[Bochner Formula For Harmonic Maps](/theorems/5688)* gives, for any local orthonormal frame $e_1,\dots,e_m$, \begin{align*} 0=(m-1)\sum_{i=1}^m |du(e_i)|^2-\sum_{i,j=1}^m h(R^N(du(e_i),du(e_j))du(e_j),du(e_i)). \end{align*} Indeed, the Ricci contribution becomes \begin{align*} \sum_{i=1}^m h(du(\operatorname{Ric}^{S^m}(e_i)),du(e_i))=\sum_{i=1}^m h(du((m-1)e_i),du(e_i))=(m-1)\sum_{i=1}^m |du(e_i)|^2. \end{align*} Since $K_N\le 0$, the sectional-curvature sign convention gives \begin{align*} -h(R^N(du(e_i),du(e_j))du(e_j),du(e_i))\ge 0 \end{align*} for every pair $i,j$. Therefore the displayed equality is a sum of nonnegative terms: \begin{align*} 0=(m-1)\sum_{i=1}^m |du(e_i)|^2+\sum_{i,j=1}^m\Bigl[-h(R^N(du(e_i),du(e_j))du(e_j),du(e_i))\Bigr]. \end{align*} Each term must vanish, so \begin{align*} (m-1)\sum_{i=1}^m |du(e_i)|^2=0. \end{align*} Because $m-1>0$, this implies $du(e_i)=0$ for every $i$, hence $du=0$ at the point. Since $\nabla du=0$, vanishing at one point forces $du=0$ everywhere on $S^m$. The sphere is connected, so $u$ is constant. This shows that the rigidity comes from the positive Ricci curvature of the domain: even a large noncompact target cannot support a nonzero parallel differential under the nonpositive target-curvature sign. [/example] ## Rigidity And Uniqueness For Negatively Curved Targets Subharmonicity controls the size of $du$, but uniqueness asks a different question: can two harmonic maps in the same homotopy class differ? Negative curvature strengthens convexity of the energy along geodesic homotopies, and this turns the variational problem into one with at most one minimising representative, modulo the flat directions allowed by geodesics. [definition: Geodesic Homotopy] Let $u_0,u_1:M\to N$ be smooth maps. A geodesic homotopy from $u_0$ to $u_1$ is a smooth map $U:M\times[0,1]\to N$ such that $U(\cdot,0)=u_0$, $U(\cdot,1)=u_1$, and for each $p\in M$, the curve $t\mapsto U(p,t)$ is a constant-speed geodesic in $N$. [/definition] When $N$ has nonpositive sectional curvature, geodesic homotopies behave like straight-line interpolations in a convex function space. To turn that analogy into a uniqueness result, we need the second variation of energy along such a homotopy to have a sign. [quotetheorem:5692] [citeproof:5692] Convexity alone says that two critical endpoints are joined by a flat segment of the energy functional. The compact-without-boundary hypothesis is what removes boundary terms in the second variation; on a manifold with boundary, the same conclusion requires fixed boundary values or another condition that kills the boundary contribution. The curvature hypothesis is also essential to this argument: for positive target curvature, the curvature term in the second variation can be negative, so energy along a geodesic homotopy need not be convex. The uniqueness question becomes geometric: what does equality in the second variation force when the target curvature is strictly negative? [quotetheorem:5693] [citeproof:5693] The theorem explains why nonpositive curvature is a natural target condition for the existence theory developed later: it gives compactness and convexity after a harmonic representative has been found. In strictly negative curvature, hyperbolic space is the main model, and the only obstruction to uniqueness is the one-dimensional geodesic case. [example: Harmonic Maps Into Hyperbolic Space] Let $N=\mathbb H^n$ have constant sectional curvature $-1$, and let $M$ be compact, connected, and without boundary. Since $\mathbb H^n$ is complete, simply connected, and satisfies $K_N=-1\le 0$, *Hartman Uniqueness Theorem* applies to any two homotopic smooth harmonic maps $u_0,u_1:M\to\mathbb H^n$. The pointwise geodesics from $u_0(p)$ to $u_1(p)$ therefore form a geodesic homotopy, and if the image of that homotopy is not contained in a single geodesic of $\mathbb H^n$, the theorem gives \begin{align*} u_0=u_1. \end{align*} It remains to interpret the exceptional case. Suppose the relevant image is contained in a geodesic $\gamma:\mathbb R\to\mathbb H^n$ parametrized by arclength. Then $|\gamma'(s)|=1$ and $\nabla^{\mathbb H^n}_{\gamma'}\gamma'=0$, and each map has the form \begin{align*} u_\ell=\gamma\circ f_\ell,\qquad \ell=0,1, \end{align*} for a smooth function $f_\ell:M\to\mathbb R$. If $e_1,\dots,e_m$ is a local $g$-orthonormal frame on $M$, the chain rule gives \begin{align*} du_\ell(e_i)=e_i(f_\ell)\gamma'(f_\ell). \end{align*} Using $|\gamma'|=1$, this implies \begin{align*} |du_\ell(e_i)|^2=e_i(f_\ell)^2|\gamma'(f_\ell)|^2=e_i(f_\ell)^2. \end{align*} Thus the energy density is exactly the real-valued energy density of $f_\ell$: \begin{align*} e(u_\ell)=\frac12\sum_{i=1}^m |du_\ell(e_i)|^2=\frac12\sum_{i=1}^m e_i(f_\ell)^2=\frac12|df_\ell|^2. \end{align*} The tension field also reduces to the scalar Laplacian. For each $i$, \begin{align*} \nabla^{\mathbb H^n}_{e_i}(du_\ell(e_i))=e_i(e_i(f_\ell))\gamma'(f_\ell)+e_i(f_\ell)^2\nabla^{\mathbb H^n}_{\gamma'}\gamma'. \end{align*} Since $\gamma$ is a geodesic, $\nabla^{\mathbb H^n}_{\gamma'}\gamma'=0$, so \begin{align*} \nabla^{\mathbb H^n}_{e_i}(du_\ell(e_i))=e_i(e_i(f_\ell))\gamma'(f_\ell). \end{align*} Also, \begin{align*} du_\ell(\nabla^M_{e_i}e_i)=(\nabla^M_{e_i}e_i)(f_\ell)\gamma'(f_\ell). \end{align*} Taking the trace in the definition of $\tau(u_\ell)$ gives \begin{align*} \tau(u_\ell)=\sum_{i=1}^m\left(e_i(e_i(f_\ell))-(\nabla^M_{e_i}e_i)(f_\ell)\right)\gamma'(f_\ell)=(\Delta_g f_\ell)\gamma'(f_\ell). \end{align*} Therefore $u_\ell$ is harmonic exactly when $f_\ell$ is harmonic. In the geodesic-image case, the problem is the ordinary theory of harmonic real-valued functions along an isometric copy of $\mathbb R$; outside that one-dimensional exception, negative curvature forces uniqueness. [/example] Flat targets sit at the boundary between rigidity and flexibility. The curvature terms vanish, so the Bochner identity says that a harmonic map has parallel differential under nonnegative Ricci curvature, but it does not force that differential to vanish without a positive Ricci input. [example: Harmonic Maps Into Flat Tori] Let $N=\mathbb R^k/\Lambda$ be a flat torus and let $M$ be compact without boundary with $\operatorname{Ric}^M\ge 0$. If $u:M\to N$ is harmonic, then the curvature tensor of $N$ vanishes, so the Bochner formula becomes \begin{align*} \Delta_g e(u)=|\nabla du|^2+\sum_{i=1}^m h(du(\operatorname{Ric}^M(e_i)),du(e_i)). \end{align*} At a fixed point, choose a $g$-orthonormal basis diagonalizing the Ricci endomorphism, so $\operatorname{Ric}^M(e_i)=\lambda_i e_i$ with $\lambda_i\ge 0$. Then linearity of $du$ in its tangent-vector input gives \begin{align*} h(du(\operatorname{Ric}^M(e_i)),du(e_i))=h(du(\lambda_i e_i),du(e_i))=\lambda_i h(du(e_i),du(e_i))=\lambda_i |du(e_i)|^2. \end{align*} Summing over $i$ therefore gives \begin{align*} \sum_{i=1}^m h(du(\operatorname{Ric}^M(e_i)),du(e_i))=\sum_{i=1}^m \lambda_i |du(e_i)|^2\ge 0. \end{align*} Hence \begin{align*} \Delta_g e(u)\ge |\nabla du|^2\ge 0. \end{align*} Integrating over the compact boundaryless manifold $M$ gives \begin{align*} 0=\int_M \Delta_g e(u)\,d\operatorname{vol}_g. \end{align*} Using the displayed Bochner identity inside the integral gives \begin{align*} 0=\int_M |\nabla du|^2\,d\operatorname{vol}_g+\int_M \sum_{i=1}^m h(du(\operatorname{Ric}^M(e_i)),du(e_i))\,d\operatorname{vol}_g. \end{align*} Both integrands are nonnegative, so the first integral can vanish only if $|\nabla du|^2=0$ everywhere by continuity. Thus every harmonic map from such an $M$ into a flat torus has parallel differential. Nonconstant examples occur when the domain is also flat. Let $M=\mathbb R^m/\Gamma$ and $N=\mathbb R^k/\Lambda$, and let $A:\mathbb R^m\to\mathbb R^k$ be a nonzero linear map with $A(\Gamma)\subset\Lambda$. Define \begin{align*} u([x])=[Ax]. \end{align*} This is well-defined because if $[x]=[y]$ in $\mathbb R^m/\Gamma$, then $x-y\in\Gamma$, so \begin{align*} Ax-Ay=A(x-y)\in\Lambda, \end{align*} and hence $[Ax]=[Ay]$ in $\mathbb R^k/\Lambda$. In lifted coordinates $\tilde u(x)=Ax$, write $A=(A_{\alpha j})$. Then \begin{align*} \tilde u_\alpha(x)=\sum_{j=1}^m A_{\alpha j}x_j. \end{align*} For each coordinate vector $\partial_i$ on $\mathbb R^m$, \begin{align*} \partial_i(\tilde u_\alpha)=A_{\alpha i}. \end{align*} Differentiating once more gives \begin{align*} \partial_i\partial_i(\tilde u_\alpha)=0. \end{align*} Therefore \begin{align*} \Delta \tilde u_\alpha=\sum_{i=1}^m \partial_i\partial_i(\tilde u_\alpha)=0 \end{align*} for every $\alpha$, so the lifted component functions are harmonic and $u$ is harmonic. Its differential is the constant linear map $A$, so $\nabla du=0$, and $u$ is nonconstant exactly when $A\neq 0$. This shows the sharp boundary of the rigidity statement: zero target curvature and merely nonnegative domain Ricci curvature allow nonzero parallel differentials, while positive Ricci curvature on the domain is what forces them to vanish. [/example] The chapter therefore leaves us with three related consequences of the same identity. The Bochner formula translates harmonicity into a scalar curvature inequality, the Eells-Sampson estimate turns curvature signs into subharmonicity, and Hartman's theorem uses the same sign condition as convexity of the energy. These tools will be reused in the regularity and existence arguments, where curvature bounds replace linear structure in controlling nonlinear elliptic systems. The Bochner identities convert geometric curvature assumptions into analytic control, giving vanishing and rigidity results for harmonic maps. Those same curvature ideas are then needed in the weak setting, where one must recover regularity from minimizers or critical points that are not assumed smooth a priori. # 9. Weak Harmonic Maps And Regularity Weak harmonic maps arise when the Dirichlet energy is minimized or varied among maps whose values are constrained to lie in a curved target. The preceding chapters developed the smooth Euler--Lagrange equation and its Bochner consequences; this chapter asks what remains true when the map is only in $W^{1,2}$. The central point is that the equation is extrinsic and nonlinear, but its variational structure still gives monotonicity, compactness, and a precise description of the defects that can occur in dimension two. ## Weak Formulation For Maps Into A Compact Target The first problem is to state the harmonic map equation for a Sobolev map whose pointwise derivatives exist only weakly. If the target is embedded in Euclidean space, admissible variations must stay tangent to the target, so the equation is not the unconstrained vector Laplace equation. Let $(M,g)$ be a smooth Riemannian domain, let $(N,h)$ be a compact Riemannian manifold, and fix an isometric embedding $N \hookrightarrow \mathbb R^q$. A map $u:M\to N$ is treated as an $\mathbb R^q$-valued Sobolev map with the constraint $u(x)\in N$ for a.e. $x$. [definition: Sobolev Map Into A Compact Target] Let $U\subset M$ be open. A map $u:U\to N$ belongs to $W^{1,2}(U;N)$ if, after the fixed isometric embedding $N\hookrightarrow\mathbb R^q$, the coordinate map $u:U\to \mathbb R^q$ belongs to $W^{1,2}(U;\mathbb R^q)$ and $u(x)\in N$ for $\mathcal L^m$-a.e. $x\in U$. [/definition] The definition is independent of the chosen compact isometric embedding up to equivalent Sobolev structures. The compactness of $N$ removes growth issues and lets us use smooth nearest-point projection on a tubular neighbourhood when constructing variations. [definition: Dirichlet Energy For A Weak Map] For an open set $U\subset M$, the Dirichlet energy on $U$ is the functional $E(\cdot;U):W^{1,2}(U;N)\to \mathbb R$ defined by \begin{align*} E(u;U)=\frac{1}{2}\int_U |du|_{g,h}^2\,d\operatorname{vol}_g. \end{align*} [/definition] This energy is the same functional as in the smooth theory, but now it is interpreted through weak first derivatives. The admissible competitors are Sobolev maps into $N$, so testing stationarity requires variations generated by tangent vector fields along $u$. [definition: Weakly Harmonic Map] A map $u\in W^{1,2}(U;N)$ is weakly harmonic if for every $\xi\in C_c^\infty(U;\mathbb R^q)$ with $\xi(x)\in T_{u(x)}N$ for a.e. $x$, the first variation vanishes: \begin{align*} \int_U \langle du,d\xi\rangle\,d\operatorname{vol}_g=0. \end{align*} [/definition] The tangent condition is often inconvenient because $T_{u(x)}N$ varies with $x$, and regularity arguments need an equation that can be tested against fixed Euclidean fields. The second fundamental form supplies exactly the missing normal correction, turning the constrained variational condition into a weak elliptic system with a quadratic right-hand side. [quotetheorem:5687] [citeproof:5687] This formulation explains why weak harmonic maps are elliptic but not automatically smooth. Compactness of $N$ is used to keep the second fundamental form and tubular projection uniformly controlled; without an embedded compact target, the same displayed equation may not define a globally bounded nonlinear term. The hypothesis $du\in L^2$ leaves $A(u)(du,du)$ only in $L^1$, so Calderon--Zygmund estimates do not directly give continuity. The theorem therefore gives the correct weak Euler--Lagrange equation, but the later regularity results need extra structure: minimizing or stationary domain variations, scale-invariant small energy, and in dimension two conformal compensation phenomena. [example: Equator-Valued Weak Harmonic Maps] Let $\Omega\subset\mathbb R^2$, and suppose that on an open set $V\subset\Omega$ the map has a single-valued phase $\theta\in W^{1,2}_{\mathrm{loc}}(V)$, so \begin{align*} u=(\cos\theta,\sin\theta)=e^{i\theta}. \end{align*} Writing $iu=(-\sin\theta,\cos\theta)$, the weak chain rule gives, for each coordinate direction $x_\alpha$, \begin{align*} \partial_\alpha u=(-\sin\theta\,\partial_\alpha\theta,\cos\theta\,\partial_\alpha\theta). \end{align*} Equivalently, \begin{align*} \partial_\alpha u=(\partial_\alpha\theta)iu. \end{align*} Since $|iu|=1$, this also gives \begin{align*} |\partial_\alpha u|^2=(\partial_\alpha\theta)^2. \end{align*} Summing over $\alpha=1,2$, \begin{align*} |du|^2=|\nabla\theta|^2. \end{align*} For a smooth phase, the second derivative in the $x_\alpha$ direction is \begin{align*} \partial_\alpha\partial_\alpha u=\partial_\alpha\bigl((\partial_\alpha\theta)iu\bigr). \end{align*} Using the product rule, \begin{align*} \partial_\alpha\partial_\alpha u=(\partial_\alpha\partial_\alpha\theta)iu+(\partial_\alpha\theta)i\partial_\alpha u. \end{align*} Substituting $\partial_\alpha u=(\partial_\alpha\theta)iu$ gives \begin{align*} \partial_\alpha\partial_\alpha u=(\partial_\alpha\partial_\alpha\theta)iu+(\partial_\alpha\theta)^2i(iu). \end{align*} Since $i(iu)=-u$, this becomes \begin{align*} \partial_\alpha\partial_\alpha u=(\partial_\alpha\partial_\alpha\theta)iu-(\partial_\alpha\theta)^2u. \end{align*} Summing over $\alpha=1,2$, \begin{align*} \Delta u=(\Delta\theta)iu-|\nabla\theta|^2u. \end{align*} The same identity holds distributionally for $\theta\in W^{1,2}_{\mathrm{loc}}$ by applying the weak chain rule and testing against compactly supported smooth functions. For $S^1\subset\mathbb R^2$, the normal correction in the extrinsic equation is $|\nabla\theta|^2u$ with this sign convention. Hence \begin{align*} \Delta u+|\nabla\theta|^2u=(\Delta\theta)iu. \end{align*} Because $iu$ has unit length everywhere, the $S^1$-valued weak harmonic map equation on $V$ is therefore equivalent to \begin{align*} \Delta\theta=0 \end{align*} in the sense of distributions. Thus a local phase is weakly harmonic exactly when it is a weak scalar harmonic function, while globally the phase may fail to exist because of winding around punctures or nontrivial loops; the extrinsic equation still remains meaningful for the map $u$ itself. [/example] The example shows that weak harmonicity captures the Euler--Lagrange equation, but regularity needs more than a target variation identity. Domain variations and comparison maps provide the monotonicity and decay estimates used below, so we separate minimizers from merely stationary weak solutions. [definition: Energy-Minimizing Harmonic Map] A map $u\in W^{1,2}(U;N)$ is energy-minimizing if for every open $V\Subset U$ and every $v\in W^{1,2}(U;N)$ with $v=u$ a.e. on $U\setminus V$, one has \begin{align*} E(u;V)\le E(v;V). \end{align*} [/definition] Minimizers are weakly harmonic by target variations, and they satisfy additional inequalities from comparison maps. The regularity theory in this chapter uses those inequalities through a scale-invariant energy quantity. ## Partial Regularity And Monotonicity The next question is whether weak harmonic maps can have singularities, and if so how large the singular set can be. The answer depends sharply on dimension: the Dirichlet energy is scale-invariant in dimension two, subcritical in dimension one, and supercritical for regularity in higher dimensions. For a ball $B(x_0,r)\subset U\subset\mathbb R^m$, the rescaled energy is the function $\Theta_u(x_0,\cdot):(0,\operatorname{dist}(x_0,\partial U))\to \mathbb R$ defined by \begin{align*} \Theta_u(x_0,r)=r^{2-m}\int_{B(x_0,r)} |du|^2\,d\mathcal L^m. \end{align*} It is designed to be unchanged by the rescaling \begin{align*} u_{x_0,r}(y)=u(x_0+ry),\qquad y\in B(0,1). \end{align*} Thus $u_{x_0,r}:B(0,1)\to N$ is the same map viewed at unit scale. Here a weak harmonic map is called stationary if the first variation of energy also vanishes under compactly supported domain diffeomorphisms. This extra hypothesis is stronger than target stationarity and is exactly what produces the stress-energy identity. [quotetheorem:5695] [citeproof:5695] The Euclidean-domain hypothesis is not cosmetic: the proof uses genuine radial vector fields and exact scaling of Lebesgue measure, while on a curved domain the formula acquires lower-order metric error terms. Stationarity is also essential; a weakly harmonic map obtained only from target variations need not have a divergence-free stress-energy tensor, so this monotonicity identity can fail. What the theorem gives is a density at every point and homogeneous tangent maps after blow-up, not smoothness by itself. The next theorem supplies the missing local criterion: if the scale-invariant energy is below a universal threshold, the weak solution enters the classical elliptic regime. [quotetheorem:5696] [citeproof:5696] The minimizing hypothesis supplies comparison maps and is stronger than weak harmonicity; for merely stationary maps, singular or discontinuous weak solutions require separate techniques and the same statement is not automatic. The smallness assumption is scale-invariant and cannot be dropped, as the map $x\mapsto x/|x|$ in dimension three has finite energy on balls but concentrates a fixed amount of rescaled energy at the origin. The theorem also gives a quantitative gradient estimate, not just continuity, which is what makes the later covering argument effective. Thus regularity becomes a question about where energy concentration persists at every scale, and we package those bad points as a singular set. [definition: Singular Set Of A Weak Harmonic Map] For an energy-minimizing harmonic map $u\in W^{1,2}(U;N)$, the singular set is \begin{align*} \operatorname{sing}(u)=\{x\in U: u\text{ is not smooth in any neighbourhood of }x\}. \end{align*} [/definition] The regular set $U\setminus\operatorname{sing}(u)$ is open by local smoothness. Epsilon regularity shows that each singular point carries a definite amount of rescaled energy at every small scale, while monotonicity gives densities that survive blow-up. The next theorem combines these two facts with dimension reduction, converting analytic concentration into a Hausdorff-dimension bound. [quotetheorem:5697] [citeproof:5697] The theorem is partial regularity rather than full regularity because higher-dimensional minimizers can genuinely be singular. Minimizing is crucial: it gives compact blow-ups that remain minimizing, whereas a general weak critical point can have worse behaviour and need not obey the same dimension-reduction argument. Compactness of $N$ again prevents the target geometry from degenerating during blow-up. The estimate $\mathcal H^{m-2}(\operatorname{sing}(u))=0$ rules out codimension-two concentration, and the sharper $m-3$ bound reflects the fact that nonconstant minimizing tangent maps do not occur in dimension two. The next example shows that isolated singularities in dimension three are not an artefact of the proof. [example: Higher-Dimensional Singular Minimizer] For $x\in B^3(0,1)\setminus\{0\}$, write $r=|x|$ and $u_i(x)=x_i/r$. Then $u$ is smooth away from $0$, and \begin{align*} |u(x)|^2=\sum_{i=1}^3\frac{x_i^2}{r^2}=\frac{r^2}{r^2}=1. \end{align*} Thus $u$ takes values in $S^2$ away from the origin. For each $i,j\in\{1,2,3\}$, \begin{align*} \partial_j u_i=\partial_j(x_i r^{-1})=\delta_{ij}r^{-1}+x_i\partial_j(r^{-1}). \end{align*} Since $\partial_j r=x_j/r$, we have \begin{align*} \partial_j(r^{-1})=-r^{-2}\partial_j r=-\frac{x_j}{r^3}. \end{align*} Therefore \begin{align*} \partial_j u_i=\frac{\delta_{ij}}{r}-\frac{x_i x_j}{r^3}=\frac{1}{r}\left(\delta_{ij}-u_i u_j\right). \end{align*} It follows that \begin{align*} |du|^2=\sum_{i,j=1}^3(\partial_j u_i)^2=\frac{1}{r^2}\sum_{i,j=1}^3(\delta_{ij}-u_i u_j)^2. \end{align*} Expanding the square gives \begin{align*} \sum_{i,j=1}^3(\delta_{ij}-u_i u_j)^2=\sum_{i,j=1}^3\delta_{ij}^2-2\sum_{i,j=1}^3\delta_{ij}u_i u_j+\sum_{i,j=1}^3u_i^2u_j^2. \end{align*} The three terms are \begin{align*} \sum_{i,j=1}^3\delta_{ij}^2=3,\qquad \sum_{i,j=1}^3\delta_{ij}u_i u_j=\sum_{i=1}^3u_i^2=1,\qquad \sum_{i,j=1}^3u_i^2u_j^2=\left(\sum_{i=1}^3u_i^2\right)^2=1. \end{align*} Hence \begin{align*} |du|^2=\frac{3-2+1}{r^2}=\frac{2}{r^2}. \end{align*} Using polar coordinates in $\mathbb R^3$, \begin{align*} \int_{B^3(0,1)}|du|^2\,d\mathcal L^3=\int_0^1\int_{S^2}\frac{2}{\rho^2}\rho^2\,d\sigma\,d\rho=2\mathcal H^2(S^2)\int_0^1d\rho=8\pi. \end{align*} Thus $du\in L^2(B^3)$, and since $u$ is bounded and $S^2$-valued a.e., we have $u\in W^{1,2}(B^3;S^2)$. The origin is a genuine singularity. Along the ray $x=te_1$ with $t>0$, one has $u(te_1)=e_1$, while along the ray $x=te_2$ one has $u(te_2)=e_2$. Since these two limiting values are different, $u(x)$ has no limit as $x\to0$ and cannot be made continuous at the origin. Its rescaled energy at the origin is constant: \begin{align*} r^{2-3}\int_{B(0,r)}|du|^2\,d\mathcal L^3=r^{-1}\int_0^r\int_{S^2}\frac{2}{\rho^2}\rho^2\,d\sigma\,d\rho=r^{-1}(8\pi r)=8\pi. \end{align*} This is why $x/|x|$ is the basic tangent model for an isolated singularity. The map is the standard energy-minimizing representative among maps with the same boundary trace, so the bound $\dim_{\mathcal H}\operatorname{sing}(u)\le m-3$ is sharp enough to allow isolated singular points when $m=3$. [/example] ## Removable Singularities In Two Dimensions Dimension two is special because the energy is conformally invariant. The main problem becomes: if a harmonic map is smooth away from an isolated point and has finite energy near that point, does the point represent a genuine singularity? [quotetheorem:5698] [citeproof:5698] The finite-energy hypothesis is the decisive two-dimensional input: without it, a punctured harmonic map may have infinite logarithmic energy and need not define a Sobolev extension across the puncture. For instance, the angular map $u:B(0,1)\setminus\{0\}\to S^1$ given by $u(re^{i\theta})=e^{i\theta}$ is smooth and harmonic away from $0$, but $|du|=1/r$ and \begin{align*} \int_{B(0,1)\setminus\{0\}} |du|^2\,d\mathcal L^2=\infty. \end{align*} It has nonzero winding around the puncture, so there is no continuous, hence no Sobolev finite-energy, extension across $0$. Compactness of $N$ keeps the image in a fixed tubular neighbourhood once local estimates are applied; noncompact targets require additional control at infinity. The theorem is local and removes only isolated punctures, so it does not prevent energy concentration in a sequence of maps. This distinction leads directly to bubbling: a single finite-energy map extends, but a bounded-energy sequence can place a whole harmonic sphere into a shrinking neighbourhood. [example: Stereographic Bubbles On The Two-Sphere] Using stereographic coordinates $z=x+iy$ centred at the concentration point, define $u_\lambda$ by the formula $u_\lambda(z)=\lambda z$. With the normalization used here, the energy density of a degree-one conformal sphere map is the pullback of the target area density \begin{align*} \frac{1}{2}|du_\lambda|^2\,d\operatorname{vol}_{S^2}=\frac{8|\partial_z(\lambda z)|^2}{(1+|\lambda z|^2)^2}\,dx\,dy. \end{align*} Since $\partial_z(\lambda z)=\lambda$, this becomes \begin{align*} \frac{1}{2}|du_\lambda|^2\,d\operatorname{vol}_{S^2}=\frac{8\lambda^2}{(1+\lambda^2|z|^2)^2}\,dx\,dy. \end{align*} Therefore \begin{align*} E(u_\lambda)=\int_{\mathbb C}\frac{8\lambda^2}{(1+\lambda^2|z|^2)^2}\,dx\,dy. \end{align*} Passing to polar coordinates $z=\rho e^{i\phi}$ gives \begin{align*} E(u_\lambda)=\int_0^\infty\int_0^{2\pi}\frac{8\lambda^2}{(1+\lambda^2\rho^2)^2}\rho\,d\phi\,d\rho. \end{align*} The angular integral contributes $2\pi$, so \begin{align*} E(u_\lambda)=16\pi\lambda^2\int_0^\infty\frac{\rho}{(1+\lambda^2\rho^2)^2}\,d\rho. \end{align*} With $s=\lambda\rho$, we have $ds=\lambda\,d\rho$ and $\rho\,d\rho=s\,ds/\lambda^2$, hence \begin{align*} E(u_\lambda)=16\pi\int_0^\infty\frac{s}{(1+s^2)^2}\,ds. \end{align*} Since \begin{align*} \frac{d}{ds}\left(-\frac{1}{2(1+s^2)}\right)=\frac{s}{(1+s^2)^2}, \end{align*} we get \begin{align*} \int_0^\infty\frac{s}{(1+s^2)^2}\,ds=\frac{1}{2}. \end{align*} Thus \begin{align*} E(u_\lambda)=16\pi\cdot\frac{1}{2}=8\pi. \end{align*} For every fixed $z\ne0$, $|\lambda z|\to\infty$ as $\lambda\to\infty$, so $u_\lambda(z)$ converges to the stereographic point at infinity. Hence $u_\lambda$ converges locally smoothly on $S^2\setminus\{0\}$ to a constant map. The energy nevertheless remains near $z=0$: for any fixed $r>0$, \begin{align*} E(u_\lambda;\{|z|<r\})=16\pi\lambda^2\int_0^r\frac{\rho}{(1+\lambda^2\rho^2)^2}\,d\rho. \end{align*} Using again $s=\lambda\rho$ gives \begin{align*} E(u_\lambda;\{|z|<r\})=16\pi\int_0^{\lambda r}\frac{s}{(1+s^2)^2}\,ds. \end{align*} Evaluating the antiderivative, \begin{align*} E(u_\lambda;\{|z|<r\})=8\pi\left(1-\frac{1}{1+\lambda^2r^2}\right). \end{align*} As $\lambda\to\infty$, the factor $1/(1+\lambda^2r^2)$ tends to $0$, so \begin{align*} E(u_\lambda;\{|z|<r\})\to8\pi. \end{align*} Thus all the energy concentrates at the point $z=0$, even though the unrescaled maps converge away from that point to a constant. If we zoom in at the concentrating scale by writing $w=\lambda z$, then \begin{align*} u_\lambda\left(\frac{w}{\lambda}\right)=w. \end{align*} The rescaled maps therefore recover the original nonconstant degree-one sphere map, which is the bubble lost in the unrescaled limit. [/example] The example illustrates the compactness failure that is unique to the critical dimension. Energy does not vanish; it moves into a different scale. ## Bubbling And Energy Quantization The final question is how to describe all possible loss of compactness for bounded-energy harmonic maps from a surface. The answer is that energy concentrates in finitely many bubbles, each bubble is a nonconstant harmonic sphere, and the neck regions between the base map and bubbles carry no energy in the limit. [definition: Bubble For A Harmonic Map Sequence] Let $(u_k)$ be a sequence of harmonic maps from a surface $\Sigma$ to a compact target $N$ with uniformly bounded energy. A bubble at a concentration point $p\in\Sigma$ is a nonconstant harmonic map $\omega:S^2\to N$ obtained as a weak limit of rescaled maps \begin{align*} x\mapsto u_k(\exp_p(r_k x)) \end{align*} for some scales $r_k\downarrow 0$. For each fixed $R<\infty$ and all sufficiently large $k$, this rescaled map is defined on $B(0,R)\subset T_p\Sigma\simeq\mathbb R^2$ whenever $\exp_p(r_kB(0,R))$ lies in the coordinate ball; the limiting map on $\mathbb R^2$ extends across infinity to $\omega:S^2\to N$ by removable singularity. [/definition] A bubble records the energy profile seen under a microscope at the concentration point. If energy remains in annular regions between two scales, those regions are called necks and require separate analysis. [definition: Neck Region] Given scales $0<r_k\ll R_k\downarrow 0$ around a concentration point $p$, the associated neck region is the annulus \begin{align*} B(p,R_k)\setminus B(p,r_k). \end{align*} [/definition] Neck regions are where the bubble-tree picture could fail: energy might spread across infinitely many intermediate scales or create a connecting curve of positive length in the target. To obtain a compactness theorem with an exact energy identity, we need to know that all energy is accounted for by the base map and finitely many bubbles. The bubble-tree compactness theorem for this setting gives an exact energy identity once the no-neck estimate is available for the target class. After passing to a subsequence, energy can be lost only at finitely many concentration points; at each such point, finitely many nonconstant harmonic spheres split off, and the total limiting energy is the energy of the weak base map plus the energies of those bubbles. Energy quantization is the two-dimensional replacement for compactness, but the stated hypotheses matter. The closed-surface assumption avoids boundary bubbles, smooth harmonicity supplies the elliptic estimates used on rescalings, and the real-analytic/no-neck input rules out hidden energy spread across long annuli. The conclusion does not say that $u_k$ converges strongly in $W^{1,2}$ on all of $\Sigma$; it says exactly where strong compactness fails and accounts for the missing energy. The defect is geometric and discrete: a finite collection of harmonic spheres splits off. This is also where topology enters the compactness theorem. A bounded-energy sequence may remain in a fixed homotopy class on $\Sigma$, but the weak limit can lose part of that class because small spheres have collapsed to concentration points. The missing topological charge is carried by the harmonic spheres $\omega_{j\ell}$, while the no-neck condition says that no additional homotopy or energy is hidden in the annular transition regions. Thus bubbling links the analytic defect measure to the target's supply of nonconstant harmonic two-spheres. [example: Neck Regions In A Bubbling Sequence] For the stereographic family $u_\lambda(z)=\lambda z$, take the annulus \begin{align*} A_{\lambda,R,r}=\{z:\lambda^{-1}R<|z|<r\}, \end{align*} where $R>0$, $r>0$, and $\lambda r>R$. From the energy-density computation for the degree-one stereographic sphere map, \begin{align*} \frac{1}{2}|du_\lambda|^2\,d\operatorname{vol}_{S^2}=\frac{8\lambda^2}{(1+\lambda^2|z|^2)^2}\,dx\,dy. \end{align*} Using polar coordinates $z=\rho e^{i\phi}$, the energy on the annulus is \begin{align*} E(u_\lambda;A_{\lambda,R,r})=\int_{\lambda^{-1}R}^{r}\int_0^{2\pi}\frac{8\lambda^2}{(1+\lambda^2\rho^2)^2}\rho\,d\phi\,d\rho. \end{align*} The angular integral contributes $2\pi$, so \begin{align*} E(u_\lambda;A_{\lambda,R,r})=16\pi\lambda^2\int_{\lambda^{-1}R}^{r}\frac{\rho}{(1+\lambda^2\rho^2)^2}\,d\rho. \end{align*} Set $s=\lambda\rho$. Then $ds=\lambda\,d\rho$, $\rho\,d\rho=s\,ds/\lambda^2$, and the endpoints become $s=R$ and $s=\lambda r$. Hence \begin{align*} E(u_\lambda;A_{\lambda,R,r})=16\pi\int_R^{\lambda r}\frac{s}{(1+s^2)^2}\,ds. \end{align*} Since \begin{align*} \frac{d}{ds}\left(-\frac{1}{2(1+s^2)}\right)=\frac{s}{(1+s^2)^2}, \end{align*} we obtain \begin{align*} E(u_\lambda;A_{\lambda,R,r})=16\pi\left(-\frac{1}{2(1+\lambda^2r^2)}+\frac{1}{2(1+R^2)}\right). \end{align*} Therefore \begin{align*} E(u_\lambda;A_{\lambda,R,r})=8\pi\left(\frac{1}{1+R^2}-\frac{1}{1+\lambda^2r^2}\right). \end{align*} For fixed $R$ and along any choice of $r=r(\lambda)$ with $\lambda r(\lambda)\to\infty$, \begin{align*} \frac{1}{1+\lambda^2r(\lambda)^2}\to0. \end{align*} Thus \begin{align*} \lim_{\lambda\to\infty}E(u_\lambda;A_{\lambda,R,r(\lambda)})=\frac{8\pi}{1+R^2}. \end{align*} Letting $R\to\infty$ gives \begin{align*} \frac{8\pi}{1+R^2}\to0. \end{align*} So the annular region between the bubble scale $\lambda^{-1}R$ and the outer scale $r(\lambda)$ carries no energy in the neck limit. In this model sequence, all the lost energy is accounted for by the bubble itself, not by the transition region between the limiting constant map and the rescaled sphere. [/example] The chapter therefore ends with a compactness picture parallel to minimal surface compactness, but with a different defect: instead of varifold multiplicity or curvature concentration along a set, two-dimensional harmonic maps lose compactness by bubbling off harmonic spheres. Weak harmonic maps introduce the existence problem in its natural compactness form: solutions may only be Sobolev, and their failures of compactness are often captured by bubbling. The final chapter replaces elliptic existence by parabolic smoothing, using heat flow to construct harmonic maps and complete the variational story. # 10. Existence By Heat Flow And The Eells-Sampson Theorem This chapter turns the existence problem for harmonic maps into a parabolic problem. In the preceding chapters, harmonic maps appeared as critical points of the Dirichlet energy and were studied through elliptic estimates, Bochner identities, and weak compactness. The heat-flow method adds a time direction: starting from an arbitrary smooth map in a fixed homotopy class, we let the map move in the direction of steepest energy decrease and ask whether the motion exists for all time and settles to a harmonic representative. Building on the Bochner identities of Chapter 8 and the weak compactness issues of Chapter 9, the central theme is that curvature of the target controls the nonlinear terms. When the target has nonpositive sectional curvature, the Bochner formula gives a priori estimates strong enough to prevent finite-time singularities and to force subsequential convergence. The Eells-Sampson theorem is the resulting existence theorem: compact-domain homotopy classes admit [harmonic representatives](/theorems/2747) when the target is compact and nonpositively curved. ## Harmonic Map Heat Flow As Gradient Flow The first question is how to turn the variational equation for harmonic maps into an evolution equation. For functions, the [heat equation](/page/Heat%20Equation) is the gradient flow of the Dirichlet integral. For maps between manifolds, the same principle holds, but the gradient must be interpreted as a section of the pullback tangent bundle. Let $(M,g)$ be a closed Riemannian manifold and $(N,h)$ a Riemannian manifold. The Dirichlet energy is the functional \begin{align*} E:C^\infty(M,N)&\longrightarrow \mathbb R, & E[u]&=\frac{1}{2}\int_M |du|^2\,d\operatorname{vol}_g. \end{align*} Here $|du|^2$ is computed using $g$ on $T_pM$ and $h$ on $T_{u(p)}N$. The Euler-Lagrange equation from the first variation was the vanishing of the tension field. For each smooth map $u:M\to N$, the tension field is a section \begin{align*} \tau(u)\in \Gamma(u^*TN), \end{align*} so the tension-field assignment sends a map to a vector field along that map: \begin{align*} \tau:C^\infty(M,N)&\longrightarrow \bigsqcup_{u\in C^\infty(M,N)}\Gamma(u^*TN), & u&\longmapsto \operatorname{trace}_g\nabla du. \end{align*} The heat flow is obtained by forcing the time derivative to equal that same vector field along $u$. [definition: Harmonic Map Heat Flow] Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. A smooth one-parameter family of maps $u:M\times[0,T)\to N$ solves the harmonic map heat flow with initial map $u_0:M\to N$ if \begin{align*} \partial_t u = \tau(u), \end{align*} and \begin{align*} u(\cdot,0)=u_0, \end{align*} where $\partial_tu$ is viewed as a section of $u^*TN$ and $\tau(u)=\operatorname{trace}_g\nabla du$ is the tension field. [/definition] After the definition, the main point is to verify that this evolution has the variational sign we want. An equation can have the same stationary points as the harmonic map equation while moving in the wrong direction for the energy, for instance if the sign were reversed to $\partial_tu=-\tau(u)$. Without an energy identity, the parabolic equation would only be a formal analogue of the Euler-Lagrange equation; the next theorem proves that it is the genuine downward $L^2$-gradient flow. [quotetheorem:5700] [citeproof:5700] This theorem explains why the flow is the right parabolic analogue of the harmonic map equation. The energy identity gives integral control of $\partial_tu$, so any limit as $t\to\infty$ should have vanishing tension field. The hypotheses matter in the computation: if $M$ has boundary and no boundary condition is imposed, integration by parts produces a boundary term that can inject or remove energy independently of the interior equation. For nonsmooth weak solutions, differentiating the energy under the integral sign also requires justification, so the identity may first hold as an inequality or only after an approximation argument. [example: Heat Flow For Circle-Valued Maps] Let $M=S^1$ with coordinate $\theta\in\mathbb R/2\pi\mathbb Z$ and let $N=S^1\subset\mathbb C$. For a degree-$k$ solution, choose a lift \begin{align*} u(e^{i\theta},t)=e^{if(\theta,t)},\qquad f(\theta+2\pi,t)=f(\theta,t)+2\pi k. \end{align*} Differentiating $u=e^{if}$ gives \begin{align*} \partial_tu=i(\partial_tf)e^{if}. \end{align*} Also, \begin{align*} \partial_\theta u=i(\partial_\theta f)e^{if}. \end{align*} Differentiating once more in $\theta$ gives \begin{align*} \partial_{\theta\theta}u=i(\partial_{\theta\theta}f)e^{if}-(\partial_\theta f)^2e^{if}. \end{align*} At the point $e^{if}\in S^1$, the tangent line is spanned by $ie^{if}$ and the radial vector $e^{if}$ is normal. Hence the tangential part of $\partial_{\theta\theta}u$ is $i(\partial_{\theta\theta}f)e^{if}$, so \begin{align*} \tau(u)=i(\partial_{\theta\theta}f)e^{if}. \end{align*} The heat-flow equation $\partial_tu=\tau(u)$ becomes \begin{align*} i(\partial_tf)e^{if}=i(\partial_{\theta\theta}f)e^{if}. \end{align*} Multiplying both sides by $-ie^{-if}$ gives the scalar heat equation \begin{align*} \partial_tf=\partial_{\theta\theta}f. \end{align*} Write \begin{align*} f(\theta,t)=k\theta+\phi(\theta,t), \end{align*} where $\phi(\theta+2\pi,t)=\phi(\theta,t)$. Since $\partial_{\theta\theta}(k\theta)=0$, the equation for $f$ becomes \begin{align*} \partial_t\phi=\partial_{\theta\theta}\phi. \end{align*} The winding condition is preserved because \begin{align*} f(\theta+2\pi,t)-f(\theta,t)=k(\theta+2\pi)-k\theta+\phi(\theta+2\pi,t)-\phi(\theta,t). \end{align*} Using the periodicity of $\phi$, this reduces to \begin{align*} f(\theta+2\pi,t)-f(\theta,t)=2\pi k. \end{align*} The energy is \begin{align*} E[u(t)]=\frac12\int_0^{2\pi}|\partial_\theta u|^2\,d\theta. \end{align*} Since $\partial_\theta u=i(\partial_\theta f)e^{if}$ and $|ie^{if}|=1$, \begin{align*} E[u(t)]=\frac12\int_0^{2\pi}(\partial_\theta f)^2\,d\theta. \end{align*} Substituting $\partial_\theta f=k+\partial_\theta\phi$ gives \begin{align*} E[u(t)]=\frac12\int_0^{2\pi}(k+\partial_\theta\phi)^2\,d\theta. \end{align*} Expanding the square gives \begin{align*} E[u(t)]=\frac12\int_0^{2\pi}\left(k^2+2k\partial_\theta\phi+(\partial_\theta\phi)^2\right)\,d\theta. \end{align*} Integrating each term separately, \begin{align*} E[u(t)]=\pi k^2+k\bigl(\phi(2\pi,t)-\phi(0,t)\bigr)+\frac12\int_0^{2\pi}(\partial_\theta\phi)^2\,d\theta. \end{align*} The middle term is zero because $\phi$ is periodic, so \begin{align*} E[u(t)]=\pi k^2+\frac12\int_0^{2\pi}(\partial_\theta\phi)^2\,d\theta. \end{align*} Differentiating this expression in $t$ gives \begin{align*} \frac{d}{dt}E[u(t)]=\int_0^{2\pi}(\partial_\theta\phi)(\partial_\theta\partial_t\phi)\,d\theta. \end{align*} Integration by parts on $[0,2\pi]$ gives \begin{align*} \frac{d}{dt}E[u(t)]=\bigl[(\partial_\theta\phi)(\partial_t\phi)\bigr]_0^{2\pi}-\int_0^{2\pi}(\partial_{\theta\theta}\phi)(\partial_t\phi)\,d\theta. \end{align*} The boundary term vanishes because $\partial_\theta\phi$ and $\partial_t\phi$ are periodic. Using $\partial_{\theta\theta}\phi=\partial_t\phi$, we obtain \begin{align*} \frac{d}{dt}E[u(t)]=-\int_0^{2\pi}(\partial_t\phi)^2\,d\theta. \end{align*} Thus the energy decreases unless $\partial_t\phi=0$. For the periodic scalar heat equation, all nonconstant Fourier modes of $\phi$ decay, while the average value is preserved. Hence $\phi(\theta,t)$ converges to its average $\bar\phi$, and the limiting lift is \begin{align*} f_\infty(\theta)=k\theta+\bar\phi. \end{align*} The limiting map is therefore \begin{align*} u_\infty(e^{i\theta})=e^{i\bar\phi}e^{ik\theta}. \end{align*} This is the affine harmonic representative in degree $k$, and the computation shows explicitly that the heat flow smooths the representative while keeping the [winding number](/page/Winding%20Number) fixed. [/example] The next obstruction is that parabolic systems may develop singularities. Energy monotonicity alone does not control the full derivative of the map pointwise, so we need a curvature condition that improves the estimates. ## Long-Time Existence Under Nonpositive Target Curvature The problem now is to rule out finite-time breakdown. For harmonic map heat flow, singularity formation is tied to concentration of $|du|^2$. The key analytic device is a parabolic Bochner inequality whose sign becomes favourable when the target has nonpositive sectional curvature. For a solution $u:M\times[0,T)\to N$, define the energy density \begin{align*} e(u)=\frac{1}{2}|du|^2. \end{align*} The Bochner formula compares $(\partial_t-\Delta_g)e(u)$ with curvature terms from $M$ and $N$. The target curvature term has the decisive sign. [quotetheorem:5701] [citeproof:5701] The inequality is useful because it is scalar: the maximum principle applies directly to $e(u)$, even though the original equation is a nonlinear system for a map into $N$. The sign is also a genuine restriction, not a cosmetic assumption. For maps into $S^2$, the corresponding target-curvature term has the opposite sign and can feed growth of $|du|^2$, which is one analytic source of concentration and bubbling. The inequality by itself also does not prove convergence; it controls the energy density locally in time and must still be paired with continuation criteria and compactness arguments for the full flow. To turn it into existence, we need to connect the scalar maximum principle estimate for $e(u)$ to the continuation criterion for the original quasilinear system; the next theorem performs exactly that step. [quotetheorem:5702] [citeproof:5702] This is the analytic heart of the Eells-Sampson method. The flow cannot break in finite time because nonpositive target curvature prevents the energy density from self-amplifying through the curvature term. Compactness of $N$ is used to keep the image in a controlled geometric region; for noncompact targets, a solution can drift to infinity even when local estimates remain valid. Closedness of $M$ removes boundary terms and avoids imposing boundary conditions; on a manifold with boundary, Dirichlet or Neumann conditions must be added and the energy computation changes. The nonpositive curvature assumption is the sign condition that distinguishes this theorem from the positive-curvature setting, where maps $S^2\to S^2$ exhibit bubbling and finite-time singular behaviour in related heat-flow problems. [example: Hyperbolic Targets] Let $M$ be closed and let $N$ be a closed hyperbolic manifold. Since $N$ is closed, it is compact; since it is hyperbolic, every sectional curvature satisfies \begin{align*} K_N(\sigma)=-1\le 0 \end{align*} for every tangent $2$-plane $\sigma\subset T_qN$. Thus the hypotheses of *Long-Time Existence For Nonpositive Target Curvature* apply, and for every smooth initial map $u_0:M\to N$ there is a smooth solution \begin{align*} u:M\times[0,\infty)\to N,\qquad u(\cdot,0)=u_0. \end{align*} The path $t\mapsto u(\cdot,t)$ is a smooth homotopy, so every time-slice $u(\cdot,t)$ lies in the same homotopy class as $u_0$. The energy identity gives \begin{align*} \frac{d}{dt}E[u(t)] = -\int_M|\partial_tu|^2\,d\operatorname{vol}_g \le 0, \end{align*} so the flow moves through the fixed homotopy class while decreasing energy. For a closed surface domain, the large-time compactness argument then extracts a harmonic representative in that same homotopy class. Negative curvature also explains the usual uniqueness phenomenon: if two harmonic maps in the same homotopy class are joined by the pointwise geodesic homotopy in $N$, the curvature term in the second variation has the favourable sign because $K_N=-1$. Equality can persist only in the degenerate case where the relevant image directions lie along a common geodesic. Thus, when the image is not collapsed into a geodesic, hyperbolic targets are the model case where the heat flow both exists for all time and selects the harmonic representative of the homotopy class. [/example] The sign condition cannot be dropped without consequences. Positive curvature allows concentration phenomena, and in two-dimensional domains this concentration is the beginning of bubbling. [example: Positive Curvature And Bubbling] For the standard round sphere $S^2$, every sectional curvature is \begin{align*} K_{S^2}(\sigma)=1>0. \end{align*} Thus the target curvature term in the parabolic Bochner formula has the opposite sign from the nonpositive-curvature case, so it is no longer forced to be nonpositive. A concrete harmonic sphere in nonzero degree is the identity map \begin{align*} \operatorname{id}:S^2\longrightarrow S^2. \end{align*} It is harmonic because the identity is an isometry and carries the Levi-Civita connection of the domain to the Levi-Civita connection of the target. Hence its second covariant derivative vanishes, and therefore \begin{align*} \tau(\operatorname{id})=\operatorname{trace}_{g_{S^2}}\nabla d(\operatorname{id})=0. \end{align*} Its energy is \begin{align*} E[\operatorname{id}]=\frac12\int_{S^2}|d(\operatorname{id})|^2\,d\operatorname{vol}_{S^2}. \end{align*} At each point $p\in S^2$, the differential $d(\operatorname{id})_p$ is the identity map on the two-dimensional tangent plane $T_pS^2$. If $e_1,e_2$ is an orthonormal basis of $T_pS^2$, then \begin{align*} |d(\operatorname{id})|^2=|d(\operatorname{id})(e_1)|^2+|d(\operatorname{id})(e_2)|^2. \end{align*} Since $d(\operatorname{id})(e_1)=e_1$ and $d(\operatorname{id})(e_2)=e_2$, this becomes \begin{align*} |d(\operatorname{id})|^2=|e_1|^2+|e_2|^2=1+1=2. \end{align*} Substituting this into the energy formula gives \begin{align*} E[\operatorname{id}]=\frac12\int_{S^2}2\,d\operatorname{vol}_{S^2}. \end{align*} Since the unit round sphere has area $4\pi$, we obtain \begin{align*} E[\operatorname{id}]=4\pi. \end{align*} This calculation gives the basic bubble energy scale: a nonconstant harmonic sphere can carry positive energy, and the degree-one round sphere carries $4\pi$. In a concentrating sequence $u_j:S^2\to S^2$, one may have weak convergence away from a point, \begin{align*} u_j\rightharpoonup u_\ast, \end{align*} while energy near that point remains nonzero: \begin{align*} \lim_{\rho\downarrow0}\liminf_{j\to\infty}\int_{B_\rho(p)}\frac12|du_j|^2\,d\operatorname{vol}_{S^2}>0. \end{align*} After rescaling around $p$, that concentrated part can converge to a nonconstant harmonic sphere. The limiting object is therefore not only the weak base map $u_\ast$, but the base map together with one or more bubbles carrying quantized harmonic-sphere energy. This is the geometric failure mode excluded by the nonpositive curvature assumption in the *Eells-Sampson Theorem*. [/example] The existence theorem gives an eternal flow, but an eternal flow need not converge. The final step is to use the energy identity and compactness to extract a harmonic limit. ## Convergence To Harmonic Representatives The question at large time is whether the flow only decreases energy or actually finds a critical point. Since the total energy drop is finite, the time derivative has finite spacetime $L^2$ norm. This means there are times at which the tension field is small, but small tension alone is not compactness. The Eells-Sampson argument also uses a large-time compactness estimate for the nonpositively curved compact-target flow: after any fixed positive waiting time, the curvature estimates and parabolic regularity give enough uniform control on a sequence of time-translates to extract a smooth subsequential limit. With this additional compactness input, almost-harmonic slices become a genuine harmonic map. [quotetheorem:5704] [citeproof:5704] This is the variational existence theorem produced by the heat-flow method. It is weaker than full convergence of the entire path, but it is enough to solve the existence problem in each smooth homotopy class. Compactness of $N$ is essential for the subsequential compactness step; without it, maps can escape down a noncompact end rather than converge. Closedness of $M$ keeps the homotopy and energy identities free of boundary conditions, while boundary-value versions require fixing compatible data. Nonpositive curvature is what rules out the bubbling behaviour familiar from positive-curvature targets such as $S^2$. The theorem does not assert uniqueness in this general form. Stronger uniqueness statements require additional hypotheses, such as negative curvature together with restrictions excluding maps whose image lies in a geodesic. These limitations point forward to the later analysis of uniqueness, rigidity, and moduli problems in negatively curved geometry, including links with hyperbolic manifolds and Teichmuller theory. [remark: Homotopy-Class Consequence] The Eells-Sampson theorem turns a topological input into an elliptic geometric representative. Instead of minimizing energy directly over a homotopy class, which may be difficult because weak limits can lose topological information, the heat flow supplies an explicit homotopy from the initial map to the harmonic map. This is why the theorem is especially useful when the homotopy class is the main datum. [/remark] The heat-flow method should be compared with the direct method in the calculus of variations. Both start from energy control, but they handle topology in different ways. [example: Comparison With The Direct Method] Suppose $u_0:M\to N$ is fixed and set \begin{align*} \mathcal H(u_0)=\{v:M\to N \mid v \text{ is smooth and homotopic to }u_0\}. \end{align*} The direct-method approach would choose a sequence $v_j\in\mathcal H(u_0)$ such that \begin{align*} E[v_j]\longrightarrow \inf_{v\in\mathcal H(u_0)}E[v]. \end{align*} Energy boundedness gives control of \begin{align*} E[v_j]=\frac12\int_M |dv_j|^2\,d\operatorname{vol}_g, \end{align*} so, after passing to a subsequence, one may obtain weak convergence \begin{align*} v_j\rightharpoonup v_\ast \end{align*} in $W^{1,2}$. The difficulty is that weak $W^{1,2}$ convergence controls the derivatives only in an integral sense; it does not give a continuous homotopy from $u_0$ to $v_\ast$. Thus the implication \begin{align*} v_j\simeq u_0\text{ for every }j \qquad\Longrightarrow\qquad v_\ast\simeq u_0 \end{align*} is not justified by weak convergence alone. The heat-flow method keeps the homotopy visible. Let $u:M\times[0,\infty)\to N$ solve harmonic map heat flow with $u(\cdot,0)=u_0$. For each finite $T>0$, the map \begin{align*} H_T:M\times[0,1]\to N,\qquad H_T(x,s)=u(x,sT), \end{align*} is a smooth homotopy from $u_0$ to $u(\cdot,T)$, because \begin{align*} H_T(x,0)=u(x,0)=u_0(x) \end{align*} and \begin{align*} H_T(x,1)=u(x,T). \end{align*} The energy identity gives \begin{align*} \frac{d}{dt}E[u(t)] = -\int_M|\partial_tu|^2\,d\operatorname{vol}_g \le 0, \end{align*} so the flow decreases energy while staying inside the same homotopy class at every finite time. Under the compact nonpositively curved target hypotheses, the large-time compactness step in the *Eells-Sampson Theorem* gives a sequence $t_j\to\infty$ with smooth convergence \begin{align*} u(\cdot,t_j)\longrightarrow u_\infty. \end{align*} For large $j$, smooth convergence implies uniform closeness of $u(\cdot,t_j)$ to $u_\infty$. Since $N$ is compact, its injectivity radius is positive, so pointwise minimizing geodesics give a smooth homotopy from $u(\cdot,t_j)$ to $u_\infty$. Therefore \begin{align*} u_0\simeq u(\cdot,t_j)\simeq u_\infty. \end{align*} The heat-flow argument replaces the missing homotopy preservation in weak convergence by an explicit path of maps, at the cost of using parabolic estimates and the nonpositive curvature condition. [/example] The chapter therefore closes the existence circle for nonpositively curved targets. The first variation identifies harmonic maps, the heat flow decreases the Dirichlet energy, the Bochner inequality prevents finite-time singularities, and compactness at large time produces a harmonic representative in the original homotopy class. ## Connections And Further Reading The two halves of the course share a common compactness pattern. First, a variational equation converts geometry into an elliptic or parabolic system: $H=0$ for minimal submanifolds and $\tau(u)=0$ for harmonic maps. Second, a monotonicity or Bochner identity isolates the quantities that can concentrate: mass ratios, excess, curvature, or energy density. Third, blow-up analysis replaces a complicated local picture by a scale-invariant model, such as a minimizing cone, a bubble sphere, or a harmonic tangent map. This shared structure also explains the difference between the theories. Minimal surface regularity is constrained by the geometry of tangent cones and by the codimension of the surface, while harmonic map regularity is constrained by the dimension of the domain and the curvature of the target. Stability, nonpositive curvature, and minimizing hypotheses are not parallel decorations; each is the condition that gives a coercive inequality strong enough to rule out the worst concentration scenarios. Good next topics are geometric measure theory for currents and varifolds, lamination compactness for embedded minimal surfaces, two-dimensional harmonic map regularity, heat-flow convergence, and bubble-tree compactness. Together they form the standard toolkit for passing from smooth variational calculations to global existence and compactness theorems. ## References - Colding and Minicozzi, *A Course in Minimal Surfaces*. - Simon, *Lectures on Geometric Measure Theory*. - Schoen and Yau, *Lectures on Harmonic Maps*. - Jost, *Riemannian Geometry and Geometric Analysis*.