Lower bound for the first eigenvalue of a minimally embedded hypersurface in a Riemannian manifold

TL;DR

This paper establishes a lower bound for the first eigenvalue of the Laplace-Beltrami operator on minimal hypersurfaces embedded in compact Riemannian manifolds with positive Ricci curvature, advancing spectral geometry understanding.

Contribution

It introduces a new lower bound for the first eigenvalue of minimal hypersurfaces in manifolds with positive Ricci curvature, linking geometric and spectral properties.

Findings

Lower bound for the first eigenvalue established

Applicable to hypersurfaces in manifolds with positive Ricci curvature

Enhances understanding of spectral geometry of minimal hypersurfaces

Abstract

We provide a lower bound for the first eigenvalue of the Laplace-Beltrami operator on a closed orientable hypersurface minimally embedded in an orientable compact Riemannian manifold with Ricci curvature bounded below by a positive constant.