Background on Riemannian Manifolds and Minimal Immersions | Gaoming Wang

Riemannian Manifolds

Mon, 29 Jun 2026 00:00:00 +0000

Definition 1.1.1

A Riemannian manifold is a smooth manifold \(M\) equipped with a Riemannian metric \(g\), which is a smooth, positive-definite symmetric \((0,2)\)-tensor field.

We use either \(g(X,Y)\) or \(\left< X,Y \right>\) to denote the inner product of two tangent vectors \(X, Y \in TM\) with respect to the metric \(g\).

The Riemann tensor \(R=R_M\) is defined by

\[ R(X,Y)Z =-\nabla_X \nabla_Y Z + \nabla_Y \nabla_X Z + \nabla_{[X,Y]} Z, \]

Here, \(\nabla\) is the Levi-Civita connection associated with \(g\), i.e., it satisfies the following properties:

Torsion-free: \(\nabla_X Y - \nabla_Y X = [X,Y]\) for all vector fields \(X, Y\).
Metric compatibility: \(X(g(Y,Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)\) for all vector fields \(X, Y, Z\).

The convention for the \((0,4)\)-type Riemann curvature tensor is

\[ R(X,Y,Z,W)=g(R(X,Y)Z,W). \]

If the ambient manifold is Euclidean space \(\mathbb{R}^N\) with the standard metric, we write \(X\cdot Y\) for the inner product of \(X\) and \(Y\), and \(D\) for the standard connection.

Submanifolds

Mon, 29 Jun 2026 00:00:00 +0000

Definition 1.2.1

Let \((M^N, g)\) be a Riemannian manifold and \(\iota: \Sigma^n \hookrightarrow M\) be a smooth immersion. The induced metric on \(\Sigma\) is defined by

\[ \bar{g} = \iota^* g, \]

so that \(\bar{g}(X,Y) = g(d\iota(X), d\iota(Y))\) for tangent vectors \(X, Y \in T\Sigma\).

Definition 1.2.2

The vector-valued second fundamental form \(\vec{A}\) is defined by

\[ \vec{A}(X,Y) = (\nabla_X Y)^\perp, \]

where \(\nabla\) is the Levi-Civita connection of \(M\) and \((\cdot)^\perp\) denotes the projection onto the normal bundle of \(\Sigma\) in \(M\). The mean curvature vector \(\vec{H}\) is defined as the trace of \(\vec{A}\):

\[ \vec{H}=\mathrm{Tr}(\vec{A}) = \sum_{i=1}^{n} \vec{A}(e_i, e_i), \]

where \(\{e_i\}\) is an orthonormal basis of \(T\Sigma\) with respect to the induced metric \(\bar{g}\).

Definition 1.2.3

A submanifold \(\Sigma\) is called a hypersurface when \(\mathrm{codim}(\Sigma) = 1\), i.e., \(n = N-1\).

If we assume \(\Sigma\) is a two-sided hypersurface, then there exists a globally defined unit normal vector field \(\nu\) on \(\Sigma\). In this case, the second fundamental form can be expressed as a scalar-valued symmetric \((0,2)\)-tensor \(A\) defined by

\[ A(X,Y) = g(\vec{A}(X,Y), \nu) = -g(\nabla_X \nu, Y) = g(\nabla_X Y, \nu). \]

Then the mean curvature \(H=g(\vec{H}, \nu)\) is the trace of \(A\).

The eigenvalues \(\kappa_1, \ldots, \kappa_{n}\) of \(A\) are called the principal curvatures.

Gauss equation for hypersurfaces.

For a hypersurface \(\Sigma\) in a Riemannian manifold \((M^N, g)\), the Gauss equation relates the intrinsic curvature of \(\Sigma\) to the extrinsic curvature and the ambient curvature:

\[ R_\Sigma(X,Y,Z,W) = R_M(X,Y,Z,W) + A(X,Z)A(Y,W) - A(X,W)A(Y,Z), \]

First and Second Variation Formulas

Mon, 29 Jun 2026 00:00:00 +0000

We consider a variation of the submanifold \(\Sigma\) given by a family of immersions \(\iota_t: \Sigma \to M\) with \(\iota_0 = \iota\). The variation vector field is defined as

\[ V = \frac{\partial \iota_t}{\partial t}\bigg|_{t=0}. \]

The volume element of \(\Sigma_t = \iota_t(\Sigma)\) is denoted by \(d\Sigma_t\).

Suppose \(U \subset \Sigma\) is an open subset. We consider the variation of the volume functional

\[ \mathcal{A}(t)=|\iota_t(U)| = \int_{U} d\Sigma_t. \]

Proposition 1.3.1

The first variation of the volume functional is given by

\[ \delta_V(U):=\mathcal{A}'(0) = \int_{ \Sigma}\mathrm{div}^\Sigma V \, d\Sigma. \]

The second variation of the volume functional is given by

\[ \begin{aligned} \delta^{2}_V(U):={}\mathcal{A}''(0) = \int_\Sigma \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + (\mathrm{div}^\Sigma V)^{2} \\ + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right> \, d\Sigma. \end{aligned} \]

Note that we do not impose any conditions on \(V\) and \(\Sigma\) for the above formulas. In particular, \(\Sigma\) does not need to be a critical point of the volume functional, and \(V\) does not need to be a normal variation.

Suppose \(\{e_i\}\) is an orthonormal basis of \(T\Sigma\), and define \(e_i(t)=\iota_{t*} e_i\). Then the area element can be expressed as

\[ d\Sigma_t = \sqrt{\det(\left< e_i(t),e_j(t) \right>)} \, d\Sigma, \]

where \(d\Sigma\) is the area element of \(\Sigma\). We need the following lemma:

Lemma 1.3.2

We have

\[ \mathrm{det}(I+tA) = 1 + t \cdot \mathrm{tr}(A) + \frac{t^{2}}{2}(\mathrm{tr}(A)^{2} - \mathrm{tr}(A^{2})) + O(t^{3}), \]

Proof. We recall the standard formula for the determinant of a matrix \(M\):

\[ \mathrm{det}(M) = \exp(\mathrm{tr}(\log M)). \]

Applying this to \(I+tA\), we have

\[ \begin{aligned} \mathrm{det}(I+tA) &= \exp(\mathrm{tr}(\log(I+tA))) \\ &= \exp\left(\mathrm{tr}\left(tA - \frac{t^{2}}{2}A^{2} + O(t^{3})\right)\right) \\ &= \exp\left(t \cdot \mathrm{tr}(A) - \frac{t^{2}}{2} \mathrm{tr}(A^{2}) + O(t^{3})\right) \\ &= 1 + t \mathrm{tr}(A) + \frac{t^{2}}{2}(\mathrm{tr}(A)^{2} - \mathrm{tr}(A^{2})) + O(t^{3}). \end{aligned} \]

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We thus obtain the first variation formula:

\[ \begin{aligned} &\frac{d}{dt}|_{t=0} |\Sigma_t\cap U| = \frac{d}{dt}|_{t=0} \int_{\Sigma} \sqrt{\det(\left< e_i,e_j \right>)} \, d\Sigma \\ ={}& \frac{1}{2} \int_{\Sigma} \mathrm{tr}(\left< e_i',e_j \right> + \left< e_i,e_j' \right> )_{ij} \, d\Sigma = n \int_{\Sigma} H \, d\Sigma\\ ={}& \int_{\Sigma} \left< \nabla_{e_i}V,e_i \right> d\Sigma=\int_{ \Sigma} \mathrm{div}^\Sigma V d\Sigma \end{aligned} \]

For the second derivative of the area element, set \(M(t)=(\left< e_i(t),e_j(t) \right>)_{ij}\). Then

\[ \begin{aligned} &\frac{d^2}{dt^2}|_{t=0} \sqrt{\mathrm{det}(M(t))}=\frac{1}{2}\frac{(\mathrm{det}\,M(t))''}{\sqrt{\mathrm{det}\,M(t)}}-\frac{1}{4}\frac{(\mathrm{det}\,M(t))'^2}{(\mathrm{det}\,M(t))^{3/2}}\\ ={}&\frac{1}{2}(\mathrm{tr}\,M''+(\mathrm{tr}\,M')^{2}-\mathrm{tr}\,(M'^{2}))-\frac{1}{4}(\mathrm{tr}\,M')^{2}\\ ={}&\frac{1}{2}\mathrm{tr}\,M''+\frac{1}{4}(\mathrm{tr}\,M')^{2}-\frac{1}{2}\mathrm{tr}\,(M'^{2}). \end{aligned} \]

We compute each term:

\[ \begin{aligned} \frac{1}{2}\mathrm{tr}\,M''={}&\sum_{i=1}^n \left< \nabla_{V}\nabla_{V}e_i,e_i \right> + \sum_{i=1}^n \left< \nabla_{V}e_i,\nabla_{V}e_i \right>\\ ={}&\sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \left< \nabla_{e_i}\nabla_{V}V,e_i \right> + \sum_{i=1}^{n}\left< \nabla_{e_i}V,\nabla_{e_i}V \right> \\ ={}&\sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + \sum_{i=1}^{n}|\nabla_{e_i} V|^{2},\\ \frac{1}{4}(\mathrm{tr}\,M')^{2}={}&\frac{1}{4}\left(\sum_{i=1}^n \left< \nabla_{V}e_i,e_i \right> + \left< e_i,\nabla_{V}e_i \right> \right)^{2}=\left(\sum_{i=1}^n \left< \nabla_{e_i}V,e_i \right> \right)^{2}=(\mathrm{div}^\Sigma V)^{2},\\ \frac{1}{2}\mathrm{tr}\,(M'^{2})={}&\frac{1}{2}\sum_{i,j=1}^n \left( \left< \nabla_{e_i}V, e_j\right> + \left< \nabla_{e_j}V, e_i \right> \right) ^{2} = \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> ^{2} + \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>\\ ={}&\sum_{i}^n |(\nabla_{e_i}V)^\top|^{2} + \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>. \end{aligned} \]

\[ \begin{aligned} \frac{d^2}{dt^2}|_{t=0} \sqrt{\mathrm{det}(M(t))} = \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + (\mathrm{div}^\Sigma V)^{2}\\ + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>. \end{aligned} \]

Then, we have the second variation formula:

\[ \begin{aligned} &\frac{d^2}{dt^2}|_{t=0} |\Sigma_t\cap U| = \int_\Sigma \frac{d^2}{dt^2}|_{t=0} \sqrt{\mathrm{det}(M(t))} \, d\Sigma \\ ={}&\int_\Sigma \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + (\mathrm{div}^\Sigma V)^{2} \\ & + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right> \, d\Sigma. \end{aligned} \]

If \(V\) has compact support in \(U\), then by the divergence theorem

\[ \begin{aligned} &\int_\Sigma \mathrm{div}^\Sigma V \, d\Sigma = \int_{ \Sigma}\mathrm{div}^\Sigma (V^\bot+V^\top) d\Sigma\\ ={}&\int_{ \Sigma}\mathrm{div}^\Sigma V^\bot d\Sigma = \int_{ \Sigma} \left< \vec{H},V \right> d\Sigma. \end{aligned} \]

Definition 1.3.3

A submanifold \(\Sigma\) is called minimal if it is a critical point of the volume functional, i.e., \(\delta_V(U) = 0\) for all variations \(V\) with compact support in \(U\).

By the first variation formula, \(\Sigma\) is minimal if and only if \(\vec{H} = 0\).

Suppose \(\Sigma\) is minimal, and the variation vector field \(V\) is normal, i.e., \(V = V^\perp\), with compact support in \(U\). Then, we know \(\mathrm{div}^\Sigma V = 0\), and \(\int_{ \Sigma} \mathrm{div}^\Sigma \nabla_{V}V \, d\Sigma = 0\). So the second variation formula simplifies to

\[ \frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| = \int_\Sigma \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right> \, d\Sigma. \]

Furthermore, if \(\Sigma\) is a two-sided hypersurface, then we can write \(V = f\nu\) for some smooth function \(f\) with compact support in \(U\). In this case, the second variation formula becomes

\[ \frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma, \]

This formula is useful in the study of minimal hypersurfaces. For instance, it directly yields the second variation formula for minimal hypersurfaces with free boundary.

Proposition 1.3.4

Suppose \((\Sigma,\partial\Sigma)\hookrightarrow (M,\partial M)\) is a minimal hypersurface with free boundary, i.e., \(\Sigma\) meets \(\partial M\) orthogonally along \(\partial \Sigma\). Then the second variation formula for normal variations \(V = f\nu\) with compact support in \(U\) is

\[ \frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma - \int_{\partial \Sigma} A_{\partial M}(\nu,\nu)f^{2} \, d\sigma, \]

Proof. It remains only to handle the term \(\int_{ \Sigma} \mathrm{div}^\Sigma \nabla_{V}V \, d\Sigma\) in the second variation formula. By the divergence theorem, we have

\[ \int_{ \Sigma} \mathrm{div}^\Sigma \nabla_{V}V \, d\Sigma = \int_{\partial \Sigma} \left< \nabla_{V}V, \eta \right> \, d\sigma=\int_{\partial \Sigma} A_{\partial M}(\nu,\nu)f^{2} \, d\sigma, \]

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Definition 1.3.5

A minimal hypersurface \(\Sigma\) is called stable if the second variation of the volume functional is nonnegative for all variations with compact support, i.e., \(\frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| \geq 0\) whenever \(\mathrm{spt}\,V \subset U\).

If \(\Sigma\) is a two-sided stable minimal hypersurface, this is equivalent to the following stability inequality for all smooth functions \(f\) with compact support in \(U\):

\[ \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \geq 0. \]

Corollary 1.3.6

The variation of mean curvature \(H\) is given by

\[ \frac{d}{dt}H=-\Delta f - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f, \]

where \(V=f\nu\) is a normal variation with compact support in \(U\).

Proof. We consider only the normal variation \(V = f\nu\). The first variation gives

\[ \int_{ \Sigma} H f d\Sigma. \]

Hence the second variation can be expressed as

\[ \int_{ \Sigma} H \frac{d}{dt}f + \frac{d}{dt}H f + (Hf)^{2} d\Sigma. \]

Comparing this expression with the second variation formula, we conclude that

\[ \begin{aligned} \int_{ \Sigma} \frac{d}{dt}H f={}&\int_{ \Sigma} -\sum_{i=1}^{n}\left< R(V,e_i)V,e_i \right> + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>\\ ={}&\int_{ \Sigma} |\nabla f|^{2}- (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} d\Sigma\\ ={}&\int_{ \Sigma} -f\left( \Delta f + (|A|^{2} + \mathrm{Ric}(\nu,\nu))f \right) d\Sigma \end{aligned} \]

Since this holds for all \(f\) with compact support in \(U\), we have

\[ \frac{d}{dt}H =-\Delta f - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f. \]

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Define the Jacobi operator \(L\) by

\[ L = -\Delta - (|A|^{2} + \mathrm{Ric}(\nu,\nu)). \]

Then the stability condition can be written as

\[ \int_\Sigma f L f \, d\Sigma \geq 0 \quad \text{for all } f \text{ with compact support in } U. \]

Definition 1.3.7

The Morse index of a minimal hypersurface \(\Sigma\) is the dimension of the space of smooth functions \(f\) with compact support in \(U\) such that

\[ \int_\Sigma f L f \, d\Sigma \lt{} 0, \]

i.e., the maximum dimension of a subspace on which the quadratic form is negative definite.

Equivalently, the Morse index counts the number of negative eigenvalues (with multiplicity) of the Jacobi operator \(L\) under appropriate boundary conditions. A minimal hypersurface is stable if and only if its Morse index is zero.

Theorem 1.3.8

Let \(\Sigma^n\) be a complete minimal hypersurface in a Riemannian manifold \((M^{n+1}, g)\). If \(\Sigma\) has finite Morse index, then \(\Sigma\) is stable outside a compact set, i.e., there exists a compact set \(K \subset \Sigma\) such that

\[ \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \geq 0 \]

for all \(f \in C_c^\infty(\Sigma \setminus K)\).

Proof. Let \(\text{ind}(\Sigma) = m \lt{} \infty\) be the Morse index of \(\Sigma\). By definition, there exists a finite-dimensional subspace \(V \subset C_c^\infty(\Sigma)\) of dimension \(m\) such that the quadratic form

\[ Q(f) = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \]

is negative definite on \(V\), and \(Q(f) \geq 0\) for all \(f \perp V\) (in the \(L^2\) inner product sense).

Since each function \(f_i\) in a basis of \(V\) has compact support, there exists a compact set \(K \subset \Sigma\) such that \(\text{spt}(f_i) \subset K\) for all \(i = 1, \ldots, m\).

Now, for any \(f \in C_c^\infty(\Sigma \setminus K)\), we have \(\text{spt}(f) \cap K = \emptyset\). This means \(f\) is orthogonal to every basis function \(f_i\) of \(V\) (since their supports are disjoint). Therefore, \(f \perp V\), and by the definition of finite Morse index, we have

\[ Q(f) = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \geq 0. \]

This proves that \(\Sigma\) is stable outside the compact set \(K\). ◻

Properties of Minimal Submanifolds in R^N

Mon, 29 Jun 2026 00:00:00 +0000

Proposition 1.4.1

Let \(\Sigma^n\) be a submanifold in \(\mathbb{R}^N\) with position vector \(x = (x_1, \ldots, x_N)\). Then its mean curvature vector \(\vec{H}\) is given by

\[ \Delta^\Sigma x = \vec{H}. \]

Proof. Let \(\{e_1, \ldots, e_n\}\) be an orthonormal basis for \(T\Sigma\). The Laplacian of the coordinate \(x_k\) on \(\Sigma\) is given by

\[ \Delta^\Sigma x_k = \sum_{i=1}^n D_{e_i} D_{e_i} x_k = \sum_{i=1}^{n} D_{e_i} (e_i \cdot \frac{\partial }{\partial x_k}) = (D_{e_i}e_i)\cdot \frac{\partial }{\partial x_k} = \vec{H} \cdot \frac{\partial }{\partial x_k}, \]

Hence

\[ \Delta^\Sigma x = \vec{H}. \]

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Proposition 1.4.2

Let \(\Sigma^n\) be a submanifold in \(\mathbb{R}^N\) with position vector \(x = (x_1, \ldots, x_N)\). Then \(\Sigma\) is minimal if and only if each coordinate function \(x_i\) is harmonic on \(\Sigma\), i.e., \(\Delta^\Sigma x_i = 0\) for all \(i = 1, \ldots, N\).

Proposition 1.4.3

Suppose \(\Sigma^n\) is a minimal submanifold in \(\mathbb{R}^{n+1}\). Then \(\nu\cdot \frac{\partial }{\partial x_i}\) satisfies the following Jacobi equation:

\[ \Delta^\Sigma (\nu\cdot \frac{\partial }{\partial x_i}) + |A|^{2}(\nu\cdot \frac{\partial }{\partial x_i}) = 0. \]

We may write this more concisely as

\[ L\nu=0. \]

Proof. This is because the flow generated by \(\frac{\partial }{\partial x_i}\) is a family of isometries of \(\mathbb{R}^{n+1}\), so \(\frac{d}{dt}H=0\) for the corresponding variation. By the variation formula for the mean curvature, we have

\[ 0=\frac{d}{dt}H = -\Delta^\Sigma (\nu\cdot \frac{\partial }{\partial x_i}) - |A|^{2}(\nu\cdot \frac{\partial }{\partial x_i}). \]

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Monotonicity Formula

Mon, 29 Jun 2026 00:00:00 +0000

Proposition 1.5.1

Suppose \(\Sigma^n\) is a minimal submanifold in \(\mathbb{R}^N\) and \(x_0\in \mathbb{R}^N\). Then we have the following monotonicity formula:

\[ \frac{|\Sigma \cap B_\rho(x_0)|}{\rho^n} - \frac{|\Sigma \cap B_\sigma(x_0)|}{\sigma^n} = \int_{\Sigma \cap (B_\rho(x_0)\setminus B_\sigma(x_0))} \frac{|(x-x_0)^\perp|^{2}}{|x-x_0|^{n+2}} \, d\Sigma \]

Proof. Assume \(x_0=0\), and choose the following vector field:

\[ V = \begin{cases} x\left( \frac{1}{|x|^n}-\frac{1}{\rho^n} \right), & \sigma\leq |x|\leq \rho\\ x\left( \frac{1}{\sigma^n}-\frac{1}{\rho^n} \right), & |x|\lt{}\sigma\\ \end{cases} \]

\[ \begin{aligned} 0={}&\int_{ \Sigma} \mathrm{div}^\Sigma V \, d\Sigma = \int_{ B_\sigma} \frac{n}{\sigma^n} d\Sigma - \int_{ \Sigma} \frac{n}{\rho^n}d\Sigma+\int_{ B_\rho\backslash B_\sigma}\mathrm{div}^\Sigma \frac{x}{|x|^n} d\Sigma\\ ={}& \frac{n|\Sigma\cap B_\sigma|}{\sigma^n} - \frac{n|\Sigma\cap B_\rho|}{\rho^n} + \int_{ B_\rho\backslash B_\sigma} \frac{n}{|x|^n}- \frac{n|x^T|^{2}}{|x|^{n+2}} d\Sigma\\ ={}& \frac{n|\Sigma\cap B_\sigma|}{\sigma^n} - \frac{n|\Sigma\cap B_\rho|}{\rho^n} + \int_{ B_\rho\backslash B_\sigma} \frac{n|x^\perp|^{2}}{|x|^{n+2}} d\Sigma. \end{aligned} \]

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Area-Minimizing Hypersurfaces and Calibration

Mon, 29 Jun 2026 00:00:00 +0000

Definition 1.6.1

A (complete) hypersurface \(\Sigma^n\) in a Riemannian manifold \((M^N, g)\) is called (absolutely) area-minimizing if it minimizes the area functional among all hypersurfaces agreeing with \(\Sigma\) outside a compact set. More precisely, \(\Sigma\) is area-minimizing if for every compact set \(K \subset M\), we have

\[ |\Sigma \cap K| \leq |\Sigma' \cap K| \]

for all hypersurfaces \(\Sigma'\) with \(\Sigma' \setminus K = \Sigma \setminus K\).

We say \(\Sigma\) is area-minimizing in its homology class if we also require \(\Sigma'\) to be homologous to \(\Sigma\), i.e., \(\Sigma-\Sigma' = \partial \Gamma\) for some \((n+1)\)-dimensional chain \(\Gamma\) in \(M\).

One can also define area-minimizing submanifolds in other classes, e.g., homotopy classes.

Proposition 1.6.2

Every area-minimizing hypersurface is minimal, i.e., its mean curvature vector \(\vec{H} = 0\).

Proposition 1.6.3

Area-minimizing hypersurfaces are stable. That is, the second variation of the area functional is non-negative for all variations with compact support.

Definition 1.6.4

Let \((M, g)\) be a Riemannian manifold. A \(k\)-form \(\omega \in \Omega^k(M)\) is called a calibration if it satisfies:

Closedness: \(d\omega = 0\).
Comass \(\leq 1\): For every point \(p \in M\) and every unit simple \(k\)-vector \(\xi\) at \(p\), we have

\[ \omega_p(\xi) \leq 1. \]

Definition 1.6.5

An oriented \(k\)-dimensional submanifold \(\Sigma^k\) in \(M\) is calibrated by a \(k\)-form \(\omega\) if \(\iota^*\omega = d\mathcal{H}^k|_\Sigma\), where \(\iota: \Sigma \hookrightarrow M\) is the inclusion map and \(d\mathcal{H}^k|_\Sigma\) is the volume form on \(\Sigma\) induced by the Riemannian metric.

Theorem 1.6.6

If an oriented \(k\)-dimensional submanifold \(\Sigma^k\) is calibrated by a closed \(k\)-form \(\omega\), then \(\Sigma\) is area-minimizing in its homology class. More precisely, for any other oriented \(k\)-dimensional submanifold \(\Sigma'\) with \(\partial\Sigma' = \partial\Sigma\), we have

\[ |\Sigma| \leq |\Sigma'|. \]

Proof. For simplicity, we assume \(\Sigma\) is a compact submanifold with boundary. Let \(\Sigma'\) be any \(k\)-dimensional oriented surface with \(\partial\Sigma' = \partial\Sigma\). Define the \((k+1)\)-dimensional chain \(\Gamma\) such that \(\partial\Gamma = \Sigma - \Sigma'\).

So we have

\[ \int_\Sigma \omega - \int_{\Sigma'} \omega = \int_\Gamma d\omega = 0, \]

since \(\omega\) is closed. Now, we have

\[ |\Sigma|=\int_{ \Sigma} \omega = \int_{ \Sigma'} \omega \leq |\Sigma'|, \]

which shows that \(\Sigma\) minimizes area among all surfaces with the same boundary. ◻

Theorem 1.6.7

Any complex analytic variety (i.e., complex submanifold or more generally, integral current defined by a holomorphic equation) in \(\mathbb{C}^2\) is absolutely area-minimizing.

Proof. The proof uses the theory of calibrations. In \(\mathbb{C}^2\), consider the standard Kähler form

\[ \omega = \frac{i}{2} (dz_1 \wedge d\bar{z}_1 + dz_2 \wedge d\bar{z}_2)=dx_1 \wedge dy_1 + dx_2 \wedge dy_2, \]

The real \(2\)-form \(\omega\) is closed (\(d\omega = 0\)) and has comass \(1\), i.e., for any oriented \(2\)-plane \(\xi\) in \(\mathbb{C}^2\), \(\omega|_\xi \leq 1\) with equality if and only if \(\xi\) is a complex line.

Any complex curve (complex \(1\)-dimensional submanifold) \(\Sigma \subset \mathbb{C}^2\) is calibrated by \(\omega\), since the restriction of \(\omega\) to \(\Sigma\) is exactly the area form of \(\Sigma\):

\[ \omega|_\Sigma = d\mathcal{H}^2|_\Sigma. \]

By the preceding calibration argument, \(\Sigma\) is area-minimizing in its homology class. Also note that \(\mathbb{R}^4\) has trivial second homology group, so any two surfaces with the same boundary are automatically homologous.

Therefore, any complex analytic variety in \(\mathbb{C}^2\) is absolutely area-minimizing. ◻

The set \(\{ z^{2}=w^3 \}\) is a complex analytic variety in \(\mathbb{C}^2\) with an isolated singularity at the origin. It is area-minimizing, but not smooth.

Minimal Graphs

Mon, 29 Jun 2026 00:00:00 +0000

Definition 1.7.1

Let \(\Omega \subset \mathbb{R}^n\) be an open domain and \(u: \Omega \to \mathbb{R}\) be a smooth function. The graph of \(u\) is the submanifold

\[ \Sigma_u = \{(x, u(x)) : x \in \Omega\} \subset \mathbb{R}^{n+1}. \]

We say \(\Sigma_u\) is a minimal graph if it is a minimal hypersurface in \(\mathbb{R}^{n+1}\), i.e., its mean curvature vanishes identically.

The graph \(\Sigma_u\) is parametrized by the immersion \(\iota: \Omega \to \mathbb{R}^{n+1}\) given by \(\iota(x) = (x, u(x))\). The tangent vectors are

\[ \eta_i=\partial_i \iota = \partial_i + \partial_i u \, \partial_{n+1}, \quad i = 1, \ldots, n, \]

where \(\{\partial_1, \ldots, \partial_{n+1}\}\) is the standard basis of \(\mathbb{R}^{n+1}\).

Basic geometry of graphs.

The induced metric on \(\Sigma_u\) is given by

\[ g_{ij} = \delta_{ij} + \partial_i u \, \partial_j u, \]

and its determinant is

\[ \det(g_{ij}) = 1 + |\nabla u|^2, \]

where \(|\nabla u|^2 = \sum_{i=1}^n (\partial_i u)^2\). Thus, the area functional of the graph over \(\Omega\) is

\[ \mathcal{A}(u) = \int_\Omega \sqrt{1 + |\nabla u|^2} \, dx. \]

The upward-pointing unit normal to \(\Sigma_u\) is

\[ \nu = \frac{(-\nabla u, 1)}{\sqrt{1 + |\nabla u|^2}} = \frac{1}{W}(-\partial_1 u, \ldots, -\partial_n u, 1), \]

where we denote \(W = \sqrt{1 + |\nabla u|^2}\).

The second fundamental form of \(\Sigma_u\) is computed by

\[ A_{ij} = -\left\langle \partial_i \nu, \partial_j \iota \right\rangle = \frac{\partial_i \partial_j u}{W} = \frac{u_{ij}}{W}. \]

The inverse of the induced metric is

\[ g^{ij} = \delta^{ij} - \frac{\partial_i u \, \partial_j u}{W^2}. \]

Proposition 1.7.2

The mean curvature of the graph \(\Sigma_u\) is given by

\[ H = \mathrm{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = \sum_{i=1}^{n} \partial_i \left(\frac{\partial_i u}{\sqrt{1+|\nabla u|^2}}\right). \]

Proof. The mean curvature is the trace of the second fundamental form with respect to the induced metric:

\[ \begin{aligned} H = g^{ij} A_{ij} &= \left(\delta^{ij} - \frac{\partial_i u \, \partial_j u}{W^2}\right)\frac{u_{ij}}{W} \\ &= \frac{1}{W}\left(\Delta u - \frac{\partial_i u \, \partial_j u \, u_{ij}}{W^2}\right)\\ &= \frac{1}{W}\Delta u - \frac{\partial_i u \, \partial_j u \, u_{ij}}{W^3}=\mathrm{div}\left(\frac{\nabla u}{W}\right). \end{aligned} \]

◻

Definition 1.7.3

The minimal surface equation (MSE) is the quasilinear elliptic PDE

\[ \mathrm{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = 0, \]

or equivalently,

\[ (1+|\nabla u|^2)\Delta u - \sum_{i,j=1}^n \partial_i u \, \partial_j u \, \partial_i \partial_j u = 0. \]

The minimal surface equation is the Euler–Lagrange equation of the area functional

\[ \mathcal{A}(u)=\int_\Omega \sqrt{1+|\nabla u|^2}\,dx. \]

Indeed, for any compactly supported variation \(\phi \in C_c^\infty(\Omega)\),

\[ \frac{d}{dt}\bigg|_{t=0} \mathcal{A}(u+t\phi) = \int_\Omega \frac{\nabla u \cdot \nabla \phi}{\sqrt{1+|\nabla u|^2}} \, dx = -\int_\Omega \mathrm{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)\phi \, dx. \]

Hence, \(u\) is a critical point of \(\mathcal{A}\) if and only if it satisfies the minimal surface equation.

Scherk’s surface. In \(\mathbb{R}^3\), the function

\[ u(x_1, x_2) = \log\left(\frac{\cos x_1}{\cos x_2}\right) \]

defined on \(\Omega = \{|x_1| \lt{} \pi/2\} \cap \{|x_2| \lt{} \pi/2\}\) is a solution of the minimal surface equation, known as Scherk’s first surface.

Catenoid. The catenoid is a minimal surface of revolution in \(\mathbb{R}^3\) that can be locally written as a graph \(u(r) = \cosh^{-1}(r)\) for \(r \geq 1\).

The Dirichlet Problem for the Minimal Surface Equation

Mon, 29 Jun 2026 00:00:00 +0000

Definition 1.8.1

The Dirichlet problem for the minimal surface equation asks: given a bounded domain \(\Omega \subset \mathbb{R}^n\) and boundary data \(\phi \in C^0(\partial\Omega)\), find \(u \in C^2(\Omega) \cap C^0(\bar{\Omega})\) such that

\[ \begin{cases} \mathrm{div}\left(\dfrac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = 0 & \text{in } \Omega, \\ u = \phi & \text{on } \partial\Omega. \end{cases} \]

Definition 1.8.2

A bounded \(C^2\) domain \(\Omega \subset \mathbb{R}^n\) is mean convex if the mean-curvature vector of \(\partial\Omega\) points weakly into \(\Omega\). Equivalently, with the scalar convention for which Euclidean balls are mean convex, the boundary mean curvature satisfies \(H_{\partial\Omega} \geq 0\).

Theorem 1.8.3

Let \(\Omega \subset \mathbb{R}^n\) be a bounded \(C^2\) domain. Then the Dirichlet problem for the minimal surface equation has a solution \(u \in C^2(\Omega) \cap C^0(\bar{\Omega})\) for every boundary data \(\phi \in C^0(\partial\Omega)\) if and only if \(\Omega\) is mean convex. In this case the solution is unique. If, moreover, \(\partial\Omega\) and \(\phi\) are \(C^{2,\alpha}\), then the solution is \(C^{2,\alpha}\) up to the boundary by the standard boundary regularity theory for quasilinear elliptic equations.

Thus mean convexity is part of the existence theorem, not merely a technical regularity assumption. On a non-mean-convex bounded domain, arbitrary boundary data need not be solvable; Jenkins–Serrin instead prove solvability under an additional smallness condition involving \(\operatorname{osc}(\phi)\) and the first two boundary derivatives of \(\phi\).

Definition 1.8.4

For a minimal graph \(\Sigma_u = \{(x, u(x)) : x \in \Omega\}\), the direction field is defined as

\[ X = \frac{(-\nabla u, 1)}{\sqrt{1 + |\nabla u|^2}}. \]

This direction field satisfies

\[ \begin{aligned} \mathrm{div}\, X &= 0, \\ X \cdot \nu &= 1. \end{aligned} \]

where \(\nu\) is the unit normal to \(\Sigma_u\).

Theorem 1.8.5

Every minimal graph in \(\mathbb{R}^{n+1}\) is area-minimizing within the class of hypersurfaces with the same boundary.

Proof. Define the vector field

\[ V=\frac{(-\nabla u, 1)}{\sqrt{1 + |\nabla u|^2}}. \]

Then the form \(\omega = \iota_V d\mathrm{vol}\) is a calibration, and \(\Sigma_u\) is calibrated by \(\omega\). Therefore, \(\Sigma_u\) minimizes area among all hypersurfaces with the same boundary. ◻

Corollary 1.8.6

Every minimal graph in \(\mathbb{R}^{n+1}\) is stable.

Bernstein’s Theorem and Generalizations

Mon, 29 Jun 2026 00:00:00 +0000

Theorem 1.9.1

Let \(u: \mathbb{R}^2 \to \mathbb{R}\) be an entire solution of the minimal surface equation. Then \(u\) is an affine function, i.e., \(u(x_1,x_2) = ax_1 + bx_2 + c\) for some constants \(a, b, c \in \mathbb{R}\).

This result was generalized to higher dimensions:

Theorem 1.9.2

Let \(u: \mathbb{R}^n \to \mathbb{R}\) be an entire solution of the minimal surface equation.

If \(n \leq 7\), then \(u\) must be affine.
For \(n \geq 8\), there exist non-affine entire solutions (as shown by a counterexample due to Bombieri–De Giorgi–Giusti [BDGG69]).

Proof. Let

\[ \Sigma=\{(x,u(x)):x\in \mathbb{R}^2\}\subset \mathbb{R}^3 \]

be the graph of an entire solution of the minimal surface equation. By the calibration argument above, \(\Sigma\) is area-minimizing. In particular, \(\Sigma\) is stable, so for every \(\varphi\in C_c^\infty(\Sigma)\),

\[ \int_\Sigma |A|^2\varphi^2\,d\Sigma \leq \int_\Sigma |\nabla^\Sigma \varphi|^2\,d\Sigma, \]

since \(\mathrm{Ric}_{\mathbb{R}^3}=0\).

We first establish a quadratic area bound. For a.e. \(R\gt{}0\), the intersection

\[ \Gamma_R:=\Sigma\cap \partial B_R \]

is a smooth \(1\)-cycle in the sphere \(\partial B_R=S_R\). Since \(H_1(S_R)=0\), the curve \(\Gamma_R\) bounds a region \(D_R\subset S_R\). Replacing \(D_R\) by its complement if necessary, we may assume

\[ |D_R|\leq \frac{1}{2}|S_R|=2\pi R^2. \]

Since \(\Sigma\) is area-minimizing and \(\partial(\Sigma\cap B_R)=\Gamma_R=\partial D_R\), we obtain

\[ |\Sigma\cap B_R|\leq |D_R|\leq 2\pi R^2 \]

for a.e. \(R\gt{}0\).

Now choose the logarithmic cutoff

\[ \eta_R(x)= \begin{cases} 1, & |x|\leq R,\\[4pt] \dfrac{k+\log R-\log |x|}{k}, & R\lt{}|x|\lt{}e^{k}R,\\[8pt] 0, & |x|\geq e^{k}R, \end{cases} \]

Here \(|x|\) denotes the Euclidean distance to the origin in \(\mathbb{R}^3\). Since \(|\nabla^\Sigma |x||\leq 1\), on the annulus \(R\lt{}|x|\lt{}e^{k}R\) we have

\[ |\nabla^\Sigma \eta_R|\leq \frac{1}{k|x|}. \]

Applying stability with \(\varphi=\eta_R\) gives

\[ \int_\Sigma |A|^2\eta_R^2\,d\Sigma \leq \int_\Sigma |\nabla^\Sigma \eta_R|^2\,d\Sigma. \]

To estimate the right-hand side, decompose

\[ B_{e^{k}R}\setminus B_R=\bigcup_{i=0}^{k-1}\bigl(B_{e^{i+1}R}\setminus B_{e^iR}\bigr), \]

Using the area bound,

\[ \begin{aligned} \int_\Sigma |\nabla^\Sigma \eta_R|^2\,d\Sigma &\leq \frac{1}{k^{2}} \sum_{i=0}^{k-1}\int_{\Sigma\cap (B_{e^{i+1}R}\setminus B_{e^iR})} \frac{1}{|x|^2}\,d\Sigma \\ &\leq \frac{1}{k^2} \sum_{i=0}^{k-1}\frac{1}{(e^iR)^2}\, |\Sigma\cap B_{e^{i+1}R}| \\ &\leq \frac{1}{k^2} \sum_{i=0}^{k-1}\frac{1}{(e^iR)^2}\,2\pi (e^{i+1}R)^2 \\ &\leq \frac{C}{k}. \end{aligned} \]

Hence

\[ \int_{\Sigma\cap B_R} |A|^2\,d\Sigma \leq \frac{C}{k}. \]

Letting \(k\to \infty\), we conclude that \(A\equiv 0\) on \(\Sigma \cap B_R\). Since this holds for a.e. \(R\gt{}0\), we have \(A\equiv 0\) on \(\Sigma\). Therefore \(\Sigma\) is totally geodesic, hence a plane in \(\mathbb{R}^3\).

Since \(\Sigma\) is a graph over \(\mathbb{R}^2\), that plane cannot be vertical. Thus

\[ u(x_1,x_2)=ax_1+bx_2+c \]

for some constants \(a,b,c\in\mathbb{R}\). This proves the theorem. ◻

The key point in dimension \(2\) is that stability plus the quadratic area growth

\[ |\Sigma\cap B_R|\leq C R^2 \]

allows the logarithmic cutoff to force

\[ \int_\Sigma |A|^2\,d\Sigma=0. \]

This argument is specific to two dimensions and yields the classical stability proof of Bernstein’s theorem.