Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces

TL;DR

This paper establishes stability estimates for isoperimetric inequalities related to Laplace eigenvalues on surfaces, demonstrating that near-maximal eigenvalues imply measures and metrics close to the optimizers in a specific Sobolev space.

Contribution

It introduces a novel approach using eigenvalues of measures to prove sharp quantitative stability for eigenvalue inequalities on various surfaces.

Findings

Proves stability estimates for first and second Laplace eigenvalues on surfaces.

Shows that near-maximal eigenvalues imply measures are close in $W^{-1,2}$ to maximizers.

Provides examples confirming the optimality of the stability power and Sobolev space choice.

Abstract

We prove stability estimates for the isoperimetric inequalities for the first and the second nonzero Laplace eigenvalues on surfaces, both globally and in a fixed conformal class. We employ the notion of eigenvalues of measures and show that if a normalized eigenvalue is close to its maximal value, the corresponding measure must be close in the Sobolev space $W^{-1,2}$ to the set of maximizing measures. In particular, this implies a qualitative stability result: metrics almost maximizing the normalized eigenvalue must be $W^{-1,2}$-close to a maximal metric. Following this approach, we prove sharp quantitative stability of the celebrated Hersch's inequality for the first eigenvalue on the sphere, as well as of its counterpart for the second eigenvalue. Similar results are also obtained for the precise isoperimetric eigenvalue inequalities on the projective plane, torus, and Klein bottle. The square of the $W^{-1,2}$ distance to a maximizing measure in these stability estimates is controlled by the difference between the normalized eigenvalue and its maximal value, indicating that the maxima are in a sense nondegenerate. We construct examples showing that the power of the distance can not be improved, and that the choice of the Sobolev space $W^{-1,2}$ is optimal.