Lower bounds on nodal sets of eigenfunctions via the heat flow

TL;DR

This paper establishes optimal lower bounds on the size of nodal sets of Laplacian eigenfunctions on compact manifolds by comparing heat flow with an artificial diffusion process, revealing new insights into eigenfunction behavior.

Contribution

It introduces a novel method using heat flow comparison to derive sharp lower bounds on nodal sets, extending classical results and proposing new conjectures.

Findings

Proves optimal lower bounds on nodal set size

Shows nodal domains cannot be confined near flat surfaces

Introduces a new heat flow comparison technique

Abstract

We study the size of nodal sets of Laplacian eigenfunctions on compact Riemannian manifolds without boundary and recover the currently optimal lower bound by comparing the heat flow of the eigenfunction with that of an artifically constructed diffusion process. The same method should apply to a number of other questions; for example, we prove a sharp result saying that a nodal domain cannot be entirely contained in a small neighbourhood of a 'reasonably flat' surface. We expect the arising concepts to have more connections to classical theory and pose some conjectures in that direction.