A lower bound for the first eigenvalue of a minimal hypersurface in the sphere

TL;DR

This paper establishes a new lower bound for the first eigenvalue of the Laplacian on minimal hypersurfaces in spheres, improving previous estimates and characterizing cases of equality based on the second fundamental form.

Contribution

It provides a sharper lower bound for the first eigenvalue of minimal hypersurfaces in spheres, with conditions for equality, using a novel Rayleigh quotient approach.

Findings

New lower bound for () involving the second fundamental form norm.

Equality ()=m when   .

Improves previous eigenvalue estimates for minimal hypersurfaces.

Abstract

Let $\Sigma$ be a closed embedded minimal hypersurface in the unit sphere $\mathbb{S}^{m+1}$ and let $\Lambda=\max\limits_{\Sigma}|A|$ be the norm of its second fundamental form. In this work we prove that the first eigenvalue of the Laplacian of $\Sigma$ satisfies $$\lambda_1(\Sigma)> \dfrac{m}{2}+\frac{m(m+1)}{32(12\Lambda+m+11)^2+8},$$ and $\lambda_1(\Sigma)=m$, when $\Lambda\le\sqrt{m}$. In particular, this estimate improves the one obtained recently in \cite{duncan2023improved}. The proof of our main result is based on a Rayleigh quotient estimate for a harmonic extension of an eigenfunction of the Laplacian of $\Sigma$ in the spirit of \cite{choi1983first}.