The Volume of a Local Nodal Domain

TL;DR

This paper provides lower bounds on the volume of local nodal domain components within a ball on closed Riemannian manifolds or surfaces, given they penetrate sufficiently deep into the ball.

Contribution

It introduces new estimates for the volume of local nodal domains in Riemannian manifolds, extending understanding of their geometric properties.

Findings

Lower bounds for nodal domain volumes in Riemannian manifolds

Conditions under which the bounds hold based on domain depth

Applicability to both real analytic and smooth manifolds

Abstract

Let M either be a closed real analytic Riemannian manifold or a closed smooth Riemannian surface. We estimate from below the volume of a nodal domain component in an arbitrary ball provided that this component enters the ball deeply enough.