A Remark on Recent Lower Bounds for Nodal Sets

TL;DR

This paper introduces a new method for establishing power law lower bounds on the volume of nodal sets of eigenfunctions, complementing existing approaches and based on growth bounds and volume relations.

Contribution

It presents a third approach using Donnelly-Fefferman growth bounds and a growth-volume relation to derive lower bounds on nodal set sizes.

Findings

Provides a new method for lower bounds on nodal sets

Extends previous results with an alternative approach

Demonstrates the effectiveness of growth-volume relations

Abstract

Recently, Sogge-Zelditch and Colding-Minicozzi gave new power law lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for eigenfunctions and a growth vs. volume relation we proved in a previous work.