New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary

TL;DR

This paper constructs new extremal domains with small volume near boundary points in Riemannian manifolds, which are close to half balls centered at critical points of mean curvature, with orthogonal boundary intersections.

Contribution

It introduces novel examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in manifolds with boundary, focusing on small prescribed volume and boundary geometry.

Findings

Domains are close to half balls centered at critical points of mean curvature.

Boundary of these domains intersects the manifold boundary orthogonally.

Constructed examples expand understanding of extremal domain configurations.

Abstract

We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold 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.