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

Abstract

We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds of dimension $n\ge 2$, with volume close to the volume of the manifold. If the first (positive) eigenfunction ${\varphi }_0$ 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 ${\varphi }_0$ attains its maximum. If ${\varphi }_0$ is a constant function and $n\ge 4$, 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.

arXiv comment40 pages