New Examples of Extremal Domains for the First Eigenvalue of the Laplace–Beltrami Operator in a Riemannian Manifold with Boundary

Abstract

We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifolds with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.