Theorems First Variation Formulae for Minimal Immersions and Harmonic Maps Attributions

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Let $(N^n,h)$ be a Riemannian manifold.

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For the area functional, let $F:M^m \to N$ be a smooth immersion, let $\Omega \subset M$ be a relatively compact [open set](/page/Open%20Set), and let $g_F := F^*h$ be the induced Riemannian metric on $M$. Let $\mu_F$ denote the Riemannian measure associated to $g_F$. Define the second fundamental form of $F$ by

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\begin{align*} B_F(X,Y) := \left(\nabla^N_X dF(Y) - dF(\nabla^{g_F}_X Y)\right)^\perp \end{align*}

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for smooth vector fields $X,Y$ on $\Omega$, where $\perp$ denotes [orthogonal projection](/theorems/437) onto the normal bundle $\nu_FM \subset F^*TN$. Define the mean curvature vector by

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\begin{align*} H_F := \operatorname{tr}_{g_F} B_F. \end{align*}

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Let $\Phi:(-\varepsilon,\varepsilon)\times \Omega\to N$ be a smooth variation, written $\Phi(t,x)=F_t(x)$, with $F_0=F$. Assume each $F_t$ is an immersion for $|t|$ sufficiently small and that the variation has compact support in $\Omega$: there is a compact set $K\subset\Omega$ such that $F_t=F$ on $\Omega\setminus K$ for all sufficiently small $|t|$. Define the variation field $V:\Omega\to F^*TN$ by

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\begin{align*} V(x):=\left.\frac{\partial F_t(x)}{\partial t}\right|_{t=0}. \end{align*}

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The variation is called normal if $V(x)\in \nu_{F,x}M$ for every $x\in \Omega$. Then $F$ is stationary for area on $\Omega$ under all compactly supported normal variations, meaning

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\begin{align*} \left.\frac{d}{dt}\right|_{t=0}\int_\Omega 1\,d\mu_{F_t}(x)=0 \end{align*}

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for every such normal variation, if and only if $H_F=0$ on $\Omega$.

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For the Dirichlet energy, let $(M^m,g)$ be a Riemannian manifold, let $u:M\to N$ be a smooth map, and let $\Omega\subset M$ be a relatively compact open set. Let $\mu_g$ denote the Riemannian measure associated to $g$. The tension field of $u$ is

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\begin{align*} \tau_g(u):=\operatorname{tr}_g \nabla du \in \Gamma(u^*TN), \end{align*}

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where $\nabla du$ is computed using the Levi-Civita connection of $(M,g)$ and the pullback connection induced by the Levi-Civita connection of $(N,h)$. Let $\Psi:(-\varepsilon,\varepsilon)\times\Omega\to N$ be a smooth variation, written $\Psi(t,x)=u_t(x)$, with $u_0=u$. Assume the variation has compact support in $\Omega$: there is a compact set $K\subset\Omega$ such that $u_t=u$ on $\Omega\setminus K$ for all sufficiently small $|t|$. Define the variation field $W:\Omega\to u^*TN$ by

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\begin{align*} W(x):=\left.\frac{\partial u_t(x)}{\partial t}\right|_{t=0}. \end{align*}

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Then $u$ is stationary for Dirichlet energy on $\Omega$ under all compactly supported variations, meaning

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\begin{align*} \left.\frac{d}{dt}\right|_{t=0}\frac{1}{2}\int_\Omega |du_t|_{g,h}^2\,d\mu_g(x)=0 \end{align*}

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for every such variation, if and only if $\tau_g(u)=0$ on $\Omega$.

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