Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

TL;DR

This paper establishes polynomial upper bounds on the Hausdorff measure of nodal sets of Laplace eigenfunctions on smooth compact manifolds, advancing understanding of their geometric complexity in relation to eigenvalues.

Contribution

It introduces a new technique for estimating the size of nodal sets using propagation of smallness and doubling index, providing bounds dependent only on eigenvalues and manifold dimension.

Findings

Hausdorff measure of nodal sets is bounded by C λ^α with α > 1/2

Develops a propagation of smallness technique for elliptic PDE solutions

Provides local bounds on nodal set volume based on frequency and doubling index

Abstract

Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $\varphi_\lambda: \Delta_M \varphi_\lambda + \lambda \varphi_\lambda = 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $\lambda$. We show that $$H^{n-1}(\{ \varphi_\lambda=0\}) \leq C \lambda^{\alpha},$$ where $\alpha>1/2$ is a constant, which depends on $n$ only, and $C>0$ depends on $\mathbb{M}$ . This result is a consequence of our study of zero sets of harmonic functions on $C^\infty$-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index.