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]