Distribution of the nodal sets of eigenfunctions on analytic manifolds

TL;DR

This paper studies how the zero sets of Laplacian eigenfunctions are distributed on analytic manifolds, showing that under certain conditions, their hypersurface measures tend to mirror the manifold's volume form, especially on negatively curved manifolds and tori.

Contribution

It proves that eigenfunctions with small-scale equidistribution have nodal sets whose volume measures converge to the manifold's volume form, extending understanding of eigenfunction nodal set distribution.

Findings

Weak limits of nodal set measures are comparable to the volume form.

Results hold for negatively curved manifolds and tori.

Full density subsequences of eigenfunctions exhibit these properties.

Abstract

The nodal set of the Laplacian eigenfunction has co-dimension one and has finite hypersurface measure on a compact Riemannian manifold. In this paper, we investigate the distribution of the nodal sets of eigenfunctions, when the metric on the manifold is analytic. We prove that if the eigenfunctions are equidistributed at a small scale, then the weak limits of the hypersurface volume form of their nodal sets are comparable to the volume form on the manifold. In particular, on the negatively curved manifolds with analytic metric and on the tori, we show that in any eigenbasis, there is a full density subsequence of eigenfunctions such that the weak limits of the hypersurface volume form of their nodal sets are comparable to the volume form on the manifold.