Hausdorff Dimension Of Nodal Set

1. 基本性质，例子 1.1. 例子和基本性质 在这一章的第一节引入了我们的研究对象，一般是一个紧的度量空间$X$装备上了一个同胚
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:$N \phi_{\lambda} =\{x:\phi_{\lambda}(x)=0\}$
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:
$H^{n-1}(N_{\phi_{\lambda}}) \leq \frac{\lambda^2||\phi_{\lambda}||_1}{2\int_{N_{\phi_{\lambda}}} |\nabla\phi_{\lambda}|dS}$