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

Charlotte Dietze

TL;DR

This work provides a quantitative rate, in Wasserstein distance along the normal direction to the boundary, for the convergence of the average density of Laplace–Beltrami eigenfunctions on a compact manifold with a singular metric near ∂X to the uniform boundary measure. By comparing the singular metric to a separable model (Grushin-type) operator ilde{Δ}, the authors derive localisation estimates via IMS-type bounds and trace controls, then obtain explicit Weyl bounds for eigenvalue counts in boundary-adjacent regions. The main results give explicit rates A_λ, depending on the boundary singularity exponent β and dimension n, for both the first moment and higher p-th moments of F_λ, together with optimality statements in the regimes β ∈ [2/n, 6/n]. The techniques—dyadic decompositions, trace estimates of model operators, and bootstrap arguments—suggest robustness to other singular or sub-Riemannian settings and yield precise information about boundary concentration of eigenfunctions in singular geometric contexts.

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 $λ$ converges weakly to the uniform normalised measure on the boundary as $λ\to\infty$. 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.

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

TL;DR

This work provides a quantitative rate, in Wasserstein distance along the normal direction to the boundary, for the convergence of the average density of Laplace–Beltrami eigenfunctions on a compact manifold with a singular metric near ∂X to the uniform boundary measure. By comparing the singular metric to a separable model (Grushin-type) operator ilde{Δ}, the authors derive localisation estimates via IMS-type bounds and trace controls, then obtain explicit Weyl bounds for eigenvalue counts in boundary-adjacent regions. The main results give explicit rates A_λ, depending on the boundary singularity exponent β and dimension n, for both the first moment and higher p-th moments of F_λ, together with optimality statements in the regimes β ∈ [2/n, 6/n]. The techniques—dyadic decompositions, trace estimates of model operators, and bootstrap arguments—suggest robustness to other singular or sub-Riemannian settings and yield precise information about boundary concentration of eigenfunctions in singular geometric contexts.

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 converges weakly to the uniform normalised measure on the boundary as . 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.
Paper Structure (7 sections, 5 theorems, 147 equations)

This paper contains 7 sections, 5 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Localisation estimates involving a trace of $\tilde{\Delta}$
  3. Estimates for the number of eigenvalues of $\tilde{\Delta}$
  4. Estimate of the localisation error and conclusion
  5. Proof of Theorem \ref{['th:main']} for $\beta=2/n$
  6. Proof of Theorem \ref{['th:main']} for $\beta>2/n$
  7. Proof of Theorem \ref{['th:moments']}

Key Result

Theorem 1

Let $\beta\in[2/n,\infty)$ and let $F_\lambda$ be the normalised density of eigenfunctions of $\Delta$ on $X$ with eigenvalues $<\lambda$, defined in eq:F_lambda_def. Define Then there exists a constant $C>0$ and $\lambda_0>0$ such that for all $\lambda\ge\lambda_0$, In particular, we also have for a possibly different constant $C>0$ that

Theorems & Definitions (14)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Corollary 1: Estimate on the convergence speed in the Wasserstein-$p$-distance
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • proof : Proof of Theorem \ref{['th:main']} for $\beta=2/n$
  • ...and 4 more