Properties of Minimal Submanifolds in R^N

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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Gaoming Wang
Gaoming Wang
Assistant Professor

My research interests include Geometric Analysis and Partial Differential Equations.