Superlevel sets and nodal extrema of Laplace-Beltrami eigenfunctions

TL;DR

This paper provides new estimates on the volume of superlevel sets and extrema distribution of Laplace-Beltrami eigenfunctions on compact manifolds, generalizing previous results to higher dimensions.

Contribution

It introduces novel techniques using Green's functions and the Bathtub principle to extend bounds on eigenfunction extrema from surfaces to arbitrary dimensions.

Findings

Upper bounds on extrema distribution over nodal domains

Generalization of previous surface results to higher dimensions

Application of Green's function and Bathtub principle techniques

Abstract

We estimate the volume of superlevel sets of Laplace-Beltrami eigenfunctions on a compact Riemannian manifold. The proof uses the Green's function representation and the Bathtub principle. As an application, we obtain upper bounds on the distribution of the extrema of a Laplace-Beltrami eigenfunction over its nodal domains. Such bounds have been previously proved by L. Polterovich and M. Sodin in the case of compact surfaces. Our techniques allow to generalize these results to arbitrary dimensions. We also discuss a different approach to the problem based on reverse H\"older inequalities due to G. Chiti.