Stability of extremal domains for the first eigenvalue of the Laplacian operator

TL;DR

This paper investigates the stability of extremal domains for the first Laplacian eigenvalue on Riemannian manifolds, providing second variation formulas, stability criteria, and classifications in spheres.

Contribution

It introduces a second variation formula for the first eigenvalue, classifies stable extremal domains in spheres, and establishes topological bounds under curvature or volume constraints.

Findings

Classification of stable extremal domains in spheres.

Second variation formula for the first eigenvalue.

Topological bounds for stable domains in Riemannian surfaces.

Abstract

In this paper, we compute the second variation of the first Dirichlet eigenvalue on extremal domains in general Riemannian manifolds and establish a criterion for stability. We classify the stable extremal domains in the 2-sphere and higher-dimensional spheres when the boundary is minimal. Additionally, we establish topological bounds for stable domains in a general compact Riemannian surface, assuming either nonnegative total Gaussian curvature or small volume.