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

旧博客原文

原题：Hausdorff Dimension Of Nodal Set

Basic setting:
Let be a compact Riemannian manifold of dimension , let be an – normalized eigenfunction of the Laplacian:

latex N \phi_{\lambda} =\{x:\phi_{\lambda}(x)=0\}$
be its nodal hypersurface. Let denote its -dimensional Riemannian hypersurface measure. In this note we prove:
Theorem:

for and metric ,there exists a constant so that:

A crucial identity:
proof of theorem 1 is based on following identity:
theorem:
for any smooth Riemannn manifold ,we have,

moreover,,

Proof:
observed we have that,

on ,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
so we have:

Estimate hausdorff measure of nodal sets:
take in theorem 2,we have:

so to get estimate hausdorff measure of nodal sets,we need to estimate:
,$||\nabla\phi_{\lambda}||_{\infty}$ ,this two guys are easy to get good estumate….and we will get a lower bound estimate of measure of nodal set:

Estimate:
,
:

normalized norm of

:
we have a yau types gradients estimate

Estimate upper bound of measure:
to get upper bound estimate,from identity we need to estimate:,,and we will get:

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。

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。