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:

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:
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$

observed we have that,
$ M=N_{\phi_{\lambda}}^+ \cup N_{\phi_{\lambda}} \cup N_{\phi_{\lambda}}^$
on $ N_{\phi_{\lambda}}^+$,use divergence theorem:
\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
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}}$


$ ||\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}$