Extremal domains for the first eigenvalue in a general Riemannian manifold

TL;DR

This paper proves the existence of small extremal domains for the first Laplace eigenvalue in any compact Riemannian manifold, extending previous results that required scalar curvature critical points.

Contribution

It generalizes prior work by removing the need for nondegenerate scalar curvature critical points, establishing existence of extremal domains under broader conditions.

Findings

Existence of extremal domains with small prescribed volume

Generalization to arbitrary compact Riemannian manifolds

Extension of previous scalar curvature critical point results

Abstract

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the Riemannian manifold was required.