Average $L^q$ growth and nodal sets of eigenfunctions of the Laplacian on surfaces

TL;DR

This paper extends the understanding of the relationship between the local growth of Laplace eigenfunctions on surfaces and their nodal set size by analyzing a broader class of $L^q$ growth exponents, providing bounds linked to eigenfunction frequency.

Contribution

It generalizes previous results by establishing bounds on nodal set size using $L^q$ growth exponents for a wider range of q, connecting growth behavior to nodal geometry.

Findings

Nodal set size is bounded by the product of $L^q$ growth and frequency.

Results extend previous $L^ extinfty$ growth bounds to $L^q$ norms.

Discussion relates findings to Yau's conjecture.

Abstract

In a recent paper, we exhibit a link between the average local growth of Laplace eigenfunctions on surfaces and the size of their nodal set. In that paper, the average local growth is computed using the uniform - or $L^\infty$ - growth exponents on disks of wavelength radius. The purpose of this note is to prove similar results for a broader class of $L^q$ growth exponents with $q \in (1, \infty)$. More precisely, we show that the size of the nodal set is bounded above and below by the product of the average local $L^q$ growth with the frequency. We briefly discuss the relation between this new result and Yau's conjecture on the size of nodal sets.