Geometric measure theory – github-backend

旧博客原文

原题：Hausdorff Dimension Of Nodal Set

Basic setting:
Let $(M,g)$ be a compact $C^\infty$ Riemannian manifold of dimension $n$ , let $\phi_{\lambda}$ be an $L^2$ – normalized eigenfunction of the Laplacian:

$\Delta \phi_{\lambda} = −\lambda^2 \phi_{\lambda}\$ and let:$ latex N \phi_{\lambda} =\{x:\phi_{\lambda}(x)=0\}$
be its nodal hypersurface. Let $H^{n−1}(N\phi_{\lambda} )$ denote its $(n-1)$ -dimensional Riemannian hypersurface measure. In this note we prove:
Theorem:

for and $C^\infty$ metric $g$ ,there exists a constant $C_g > 0$ so that:
$H^{n-1}(N_{\phi_{\lambda}}) \leq C_g \lambda^{n}$

A crucial identity:
proof of theorem 1 is based on following identity:
theorem:
for any smooth Riemannn manifold $M$ ,we have,
$\lambda^2\int_{M}|\phi_{\lambda}|dV = 2\int_{N_{\phi_{\lambda}}} |\nabla\phi_{\lambda}|dS$
moreover, $\forall f \in C^2(M)$ ,
$\int_M(\Delta+\lambda^2)f \vert\phi_{\lambda}\vert dV=2\int_{N_{\phi_{\lambda}}} \vert\nabla\phi_{\lambda}\vert dS$

Proof:
observed we have that,
$M=N_{\phi_{\lambda}}^+ \cup N_{\phi_{\lambda}} \cup N_{\phi_{\lambda}}^$
on $N_{\phi_{\lambda}}^+$ ,use divergence theorem:
\begin{eqnarray*}
\int_M(\Delta+\lambda^2)f \phi_{\lambda} dV&=&\int_M(\Delta+\lambda^2)\phi_{\lambda}f dV+\int_{\partial M} -g(\upsilon,\phi_{\lambda}\nabla f)dS+\int_{\partial M} g(\upsilon,f\nabla\phi_{\lambda})dS\\
&=&\int_{\partial M} g(\upsilon,f\nabla\phi_{\lambda}) \\
&=&\int_{\partial M} f\phi_{\lambda}dS
\end{eqnarray*}
the same identity is true on $N_{\phi_{\lambda}}^-$
so we have:
$\int_M(\Delta+\lambda^2)f \vert\phi_{\lambda}\vert dV=2\int_{N_{\phi_{\lambda}}} \vert\nabla\phi_{\lambda}\vert dS$

Estimate hausdorff measure of nodal sets:
take $f=1$ in theorem 2,we have:
$\lambda^2\int_M\vert\phi_{\lambda}\vert dV=2\int_{N_{\phi_{\lambda}}} \vert\nabla\phi_{\lambda}\vert dS$
so to get estimate hausdorff measure of nodal sets,we need to estimate:
$||\phi_{\lambda}||_1$ ,$||\nabla\phi_{\lambda}||_{\infty}$ ,this two guys are easy to get good estumate….and we will get a lower bound estimate of measure of nodal set:
$H^{n-1}(N_{\phi_{\lambda}}) \geq \frac{\lambda^2||\phi_{\lambda}||_1}{2||\nabla\phi_{\lambda}||_{\infty}}$

Estimate:
$||\phi_{\lambda}||_1$ , $||\nabla\phi_{\lambda}||_{\infty}$
$||\phi_{\lambda}||_1$ :

normalized $L_2$ norm of $\phi_{\lambda}$

$||\nabla\phi_{\lambda}||_{\infty}$ :
we have a yau types gradients estimate

Estimate upper bound of measure:
to get upper bound estimate,from identity we need to estimate: $||\phi_{\lambda}||_1$ , $||\nabla\phi_{\lambda}||_{\infty}$ ,and we will get:
$H^{n-1}(N_{\phi_{\lambda}}) \leq \frac{\lambda^2||\phi_{\lambda}||_1}{2\int_{N_{\phi_{\lambda}}} |\nabla\phi_{\lambda}|dS}$

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

Laplace 特征函数的 nodal set 是

$$N_\lambda=\{x:\phi_\lambda(x)=0\}.$$

它的大小反映了 eigenfunction 的振荡。Yau 猜想预言在光滑紧流形上，nodal hypersurface 的测度与 $\lambda$ 同阶。

Laplace 特征函数的 nodal set：Dong identity 与 Hausdorff measure 下界 — Laplace 特征函数的 nodal set 测度可通过 Dong identity 与梯度估计联系起来。

1. 基本设定

设

$$-\Delta\phi_\lambda=\lambda^2\phi_\lambda,\qquad \|\phi_\lambda\|_2=1.$$

nodal set 通常是一个维数 $n-1$ 的几何对象，但可能带有奇异点。

2. Dong identity

一个关键恒等式是：对光滑函数 $f$，

$$\int_M(\Delta+\lambda^2)f\,|\phi_\lambda|\,dV
=2\int_{N_\lambda} f|\nabla\phi_\lambda|\,dS.$$

它把 nodal set 上的积分转成整个流形上的积分。

3. 下界策略

取 $f=1$，得到

$$\lambda^2\int_M|\phi_\lambda|\,dV
=2\int_{N_\lambda}|\nabla\phi_\lambda|\,dS.$$

于是

$$\mathcal H^{n-1}(N_\lambda)\gtrsim
\frac{\lambda^2\|\phi_\lambda\|_1}{\|\nabla\phi_\lambda\|_\infty}.$$

4. 需要的两个估计

要得到 nodal measure 下界，需要控制 $\|\phi_\lambda\|_1$ 的下界和 $\|\nabla\phi_\lambda\|_\infty$ 的上界。后者来自 elliptic estimates 或 spectral cluster estimates。

5. 几何意义

nodal set 是 eigenfunction 改变符号的地方。特征值越大，振荡越快，nodal set 应该越大。Dong identity 精确表达了这种振荡与零集几何之间的关系。

分类： Geometric measure theory

Laplace 特征函数的 nodal set：Dong identity 与 Hausdorff measure 下界