# [Lecture notes]A Course in Minimal Surface

A Course in Minimal Surface 09/10/2013 参考教材:

1. Consequence of the first variation formulae 1.1. Variational formula 1.2. Laplacian 1.3. Coarea formula 1.4. Monotonically formula 1.5. Mean value inequality 2. The Theorem of Berstein 2.1. Logarithmic cutoff trick The notes of 20130909 3. Strong Maximum Principle

Corollary 1. 假设$\Sigma_1$, $\Sigma_2$是$\R^n$中的无边界、完备、连通、极小超曲面, 若$\Sigma_1\cap\Sigma_2\neq\phi$, 且$\Sigma_1$完全位于$\Sigma_2$的一边, 则$\Sigma_1=\Sigma_2$.

4. Second Variational formula and Stability

Definition 2. $\Sigma$ is called parabolic if any positive superharmoinc function on $\Sigma$ is constant.
Proposition 3. Suppose that $\Sigma$ is a complete surface with $\text{Area}(B_s^\Sigma)\leq C s^2$ for some constant $C$, then $\Sigma$ is parabilic.
5. Local Examples of Multi-Valued graphs 5.1. Weierstrass Representation 给定一个区域$\Omega$, 以及一个$\Omega$上的亚纯函数$g$, 和一个$\Omega$桑的全纯函数$\phi$, 我们可以构造一个映射$F:\Omega\to\R^3$, 使得$F(\Omega)$作为$\R^3$的子集是极小曲面. 其中$F$为 $$F(z)=\mathrm{Re}\int_{\xi\in\gamma_{z_0},z}\left( \frac{1}{2}\left(g^{-1}(\xi)-g(\xi)\right), \frac{i}{2}\left(g^{-1}(\xi)+g(\xi)\right), 1 \right)\phi(\xi)\rd\xi$$ 这里$z_0\in\Omega$为一固定点, 而$\gamma_{z_0,z}$为连接$z_0$, $z$的连续曲线. $F$就称为Weierstrass 表示. 5.2. Definition of Multi-valued graph 假设$D_r=\set{z\in\C||z|< r }$, $P$为$\C\setminus\set{0}$的万有覆盖. 例如我们可以取$P$为有半开平面$\set{(\rho,\theta)|\rho > 0,\theta\in\R}$, 则映射$\pi:P\to\C\setminus\set{0}$, $\pi(\rho,\theta)=\rho e^{i\theta}$就是覆叠映射(无穷重的). 这里$(\rho,\theta)$称为$P$的整体极坐标.
Definition 4. A $n$-valued graph on annulus $D_s\setminus D_r\subset\C\setminus\set{0}$ is a single valued graph of a fucntion $u$ over $\set{(\rho,\theta)|r< \rho\leq s,|\theta|\leq n\pi}$.

The notes of 20130923 6. Curvature Estimate 6.1. Simon’s inequality

Theorem 5. Suppose $\Sigma^{n-1}$ be a minimal surface in $\R^n$. Then we have Simon’s inequality: $$\Delta^2|A|^2\geq-2|A|^4+2\left(1+\frac{2}{n-1})\right)|\nabla^\Sigma|A||^2.$$

Remark 1. 特别, 当$n=2$时, Simon’s 不等式变为: $$\Delta^\Sigma|A|^2=-2|A|^4+4|\nabla\Sigma|A||^2.$$
6.2. small energy curvature estimates
• Choi-Schoen关于具有”小”全曲率的极小曲面的曲率内估计.
• Heinz 关于完全极小图的曲率估计.
The notes of 20130930