Monotonicity Formula
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:
\[
V =
\begin{cases}
x\left( \frac{1}{|x|^n}-\frac{1}{\rho^n} \right), & \sigma\leq |x|\leq \rho\\
x\left( \frac{1}{\sigma^n}-\frac{1}{\rho^n} \right), & |x|\lt{}\sigma\\
\end{cases}
\]
So
\[
\begin{aligned}
0={}&\int_{ \Sigma} \mathrm{div}^\Sigma V \, d\Sigma = \int_{ B_\sigma} \frac{n}{\sigma^n} d\Sigma - \int_{ \Sigma} \frac{n}{\rho^n}d\Sigma+\int_{ B_\rho\backslash B_\sigma}\mathrm{div}^\Sigma \frac{x}{|x|^n} d\Sigma\\
={}& \frac{n|\Sigma\cap B_\sigma|}{\sigma^n} - \frac{n|\Sigma\cap B_\rho|}{\rho^n} + \int_{ B_\rho\backslash B_\sigma} \frac{n}{|x|^n}- \frac{n|x^T|^{2}}{|x|^{n+2}} d\Sigma\\
={}& \frac{n|\Sigma\cap B_\sigma|}{\sigma^n} - \frac{n|\Sigma\cap B_\rho|}{\rho^n} + \int_{ B_\rho\backslash B_\sigma} \frac{n|x^\perp|^{2}}{|x|^{n+2}} d\Sigma.
\end{aligned}
\]
◻
Gaoming Wang
Assistant Professor
My research interests include Geometric Analysis and Partial Differential Equations.