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:

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}}



normalized L_2 norm of \phi_{\lambda}

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}