Minimal Graphs | Gaoming Wang

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:

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

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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 \(r \geq 1\).

Gaoming Wang

Assistant Professor