Lower bounds for nodal sets of Dirichlet and Neumann eigenfunctions

TL;DR

This paper establishes lower bounds on the size of nodal sets for Dirichlet and Neumann eigenfunctions of the Laplace-Beltrami operator on compact Riemannian manifolds with boundary, advancing understanding of eigenfunction behavior.

Contribution

It provides new lower bounds for the measure of nodal sets of eigenfunctions with boundary conditions, a significant step in spectral geometry.

Findings

Lower bounds for nodal set size of eigenfunctions

Results apply to manifolds with boundary

Advances in spectral geometry understanding

Abstract

Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {\phi=0}.