Maximizing one Laplace eigenvalue on n-dimensional manifolds

TL;DR

This paper establishes existence and regularity results for maximizing a Laplace eigenvalue within a conformal class of metrics on n-dimensional manifolds, advancing understanding of spectral geometry.

Contribution

It provides the first rigorous proof of existence and regularity for the maximization problem of a Laplace eigenvalue in conformal classes on higher-dimensional manifolds.

Findings

Existence of maximizers for the eigenvalue problem.

Regularity properties of the maximizing metrics.

Extension of spectral optimization results to higher dimensions.

Abstract

We prove existence and regularity results for the problem of maximization of one Laplace eigenvalue with respect to metrics of same volume lying in a conformal class of a Riemannian manifold of dimension $n\geq 3$.