Nodal sets and growth exponents of Laplace eigenfunctions on surfaces

TL;DR

This paper establishes bounds connecting the local growth of Laplace eigenfunctions on surfaces to the size of their nodal sets, providing insights into eigenfunction behavior and supporting Yau's conjecture in the semi-classical limit.

Contribution

It proves bounds relating nodal set size to eigenfunction growth, linking local growth rates to global nodal set measures on surfaces.

Findings

Bounds on Hausdorff measure of nodal sets in terms of growth exponents

Relation between local growth and nodal set size supports Yau's conjecture

Extension of results to solutions of planar Schrödinger equations with small potential

Abstract

We prove a result, announced by F. Nazarov, L. Polterovich and M. Sodin that exhibits a relation between the average local growth of a Laplace eigenfunction on a closed surface and the global size of its nodal set. More precisely, we provide a lower and an upper bound to the Hausdorff measure of the nodal set in terms of the expected value of the growth exponent of an eigenfunction on disks of wavelength like radius. Combined with Yau's conjecture, the result implies that the average local growth of an eigenfunction on such disks is bounded by constants in the semi-classical limit. We also obtain results that link the size of the nodal set to the growth of solutions of planar Schr\"odinger equations with small potential.