Extremal Domains of Big Volume for the First Eigenvalue of the Laplace-Beltrami Operator in a Compact Manifold

TL;DR

This paper establishes the existence and characterization of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in compact manifolds, linking their shape to eigenfunction properties and scalar curvature.

Contribution

It proves the existence of extremal domains near geodesic balls in compact manifolds, depending on the eigenfunction's behavior and scalar curvature, extending previous spectral geometry results.

Findings

Extremal domains are close to complements of small geodesic balls.

Domains relate to maximum points of the first eigenfunction.

Results depend on whether the eigenfunction is constant or not.

Abstract

We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds of dimension $n \geq 2$, with volume close to the volume of the manifold. If the first (positive) eigenfunction $\phi_0$ of the Laplace-Beltrami operator over the manifold is a nonconstant function, these domains are close to the complement of geodesic balls of small radius whose center is close to the point where $\phi_0$ attains its maximum. If $\phi_0$ is a constant function and $n \geq 4$, these domains are close to the complement of geodesic balls of small radius whose center is close to a nondegenerate critical point of the scalar curvature function.