Convergence speed for the average density of eigenfunctions for singular Riemannian manifolds

TL;DR

This paper investigates how quickly the average density of eigenfunctions on singular Riemannian manifolds with boundary converges to a uniform measure on the boundary, providing quantitative estimates in Wasserstein distance.

Contribution

It offers a new quantitative estimate on the convergence speed of eigenfunction densities in the Wasserstein sense for singular Riemannian manifolds with boundary.

Findings

Convergence of eigenfunction densities to boundary measure as eigenvalue increases

Quantitative estimates of convergence speed in Wasserstein distance

Applicability to singular Riemannian metrics with boundary

Abstract

We consider a class of singular Riemannian metrics on a compact Riemannian manifold with boundary and the eigenfunctions of the corresponding Laplace-Beltrami operator. In our setting, the average density of eigenfunctions with eigenvalue less than $\lambda$ converges weakly to the uniform normalised measure on the boundary as $\lambda\to\infty$. In this work, we show a quantitative estimate on the speed of this convergence in the Wasserstein-sense in the transverse coordinate to the boundary.