Lower bounds on the Hausdorff measure of nodal sets

TL;DR

This paper establishes a new polynomial lower bound on the Hausdorff measure of nodal sets of Laplace eigenfunctions, improving significantly over previous exponential bounds.

Contribution

It provides the first polynomial lower bound on the measure of nodal hypersurfaces for eigenfunctions on smooth Riemannian manifolds.

Findings

Lower bound: al^{n-1}( cal_{\u03c6_{\u03bb}}) \u2265 C \u03bb^{rac74-rac{3n}4}

Improves previous exponential bounds to polynomial bounds

Applicable to eigenfunctions on smooth Riemannian manifolds

Abstract

Let $\ncal_{\phi_{\lambda}}$ be the nodal hypersurface of a $\Delta$-eigenfunction $\phi_{\lambda}$ of eigenvalue $\lambda^2$ on a smooth Riemannian manifold. We prove the following lower bound for its surface measure: $\hcal^{n-1}(\ncal_{\phi_{\lambda}}) \geq C \lambda^{\frac74-\frac{3n}4} $. The best prior lower bound appears to be $e^{- C \lambda}$.