Area-Minimizing Hypersurfaces and Calibration

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:

  1. Closedness: \(d\omega = 0\).

  2. 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.

Gaoming Wang
Gaoming Wang
Assistant Professor

My research interests include Geometric Analysis and Partial Differential Equations.