# 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: $H^{n-1}(N_{\phi_{\lambda}}) \leq \frac{\lambda^2||\phi_{\lambda}||_1}{2\int_{N_{\phi_{\lambda}}} |\nabla\phi_{\lambda}|dS}$