Laplace Eigenfunctions And Damped Wave Equation Ii: Product Manifolds

TL;DR

This paper investigates eigenfunction concentration on product manifolds and extends stabilization results for weakly damped wave equations to these settings, broadening previous work on tori.

Contribution

It applies and extends existing eigenfunction concentration methods to product manifolds, deriving new stabilization results for damped wave equations.

Findings

Eigenfunctions can concentrate in specific regions of product manifolds.

Stabilization results for weakly damped wave equations are generalized to product manifolds.

The approach builds on and extends prior methods used for tori.

Abstract

- The purpose of this article is to study possible concentrations of eigenfunc-tions of Laplace operators (or more generally quasi-modes) on product manifolds. We show that the approach of the first author and Zworski [10, 11] applies (modulo rescalling) and deduce new stabilization results for weakly damped wave equations which extend to product manifolds previous results by Leautaud-Lerner [12] obtained for products of tori.