A natural lower bound for the size of nodal sets

Abstract

We prove that, for an $n$-dimensional compact Riemannian manifold $\left(M,g\right)$, the $\left(n-1\right)$-dimensional Hausdorff measure $\left|Z_{\lambda }\right|$ of the zero-set $Z_{\lambda }$ of an eigenfunction $e_{\lambda }$ of the Laplacian having eigenvalue $-\lambda$, where $\lambda \ge 1$, and normalized by ${\int }_M|e_{\lambda }{|}^{2}d{V}_g=1$ satisfies

$$C\left|Z_{\lambda }\right|\ge {\lambda }^{\frac{1}{2}}{\left({\int }_M\left|e_{\lambda }\right|d{V}_g\right)}^2$$

for some uniform constant $C$. As a consequence, we recover the lower bound $\left|Z_{\lambda }\right|\gtrsim {\lambda }^{\left(3-n\right)∕4}$.