Variational aspects of Laplace eigenvalues on Riemannian surfaces

TL;DR

This paper investigates the maximization of Laplace eigenvalues on Riemannian surfaces within conformal classes, establishing existence, regularity, and properties of extremal metrics using variational methods.

Contribution

It introduces a general variational framework for eigenvalue maximization problems and proves existence and regularity results for extremal metrics, including a partially regular maximizer for the first eigenvalue.

Findings

Existence of maximising metrics for Laplace eigenvalues on surfaces.

Regularity properties of extremal metrics are established.

A partially regular maximizer for the first eigenvalue is proven to exist.

Abstract

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the general regularity properties of $\lambda_k$-extremal metrics and the existence of a partially regular $\lambda_1$-maximiser.