An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere

TL;DR

This paper establishes a new explicit lower bound for the first non-zero eigenvalue of the Laplace-Beltrami operator on embedded minimal hypersurfaces in spheres, improving previous bounds without additional assumptions.

Contribution

It provides the first explicit and computable improvement on the classical lower bound for the eigenvalue of minimal hypersurfaces in spheres.

Findings

Derived a lower bound involving the second fundamental form

Explicit constants for the bound are provided

Improves upon the classical bound without extra assumptions

Abstract

Suppose that $\Sigma^n\subset\mathbb{S}^{n+1}$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $\lambda_1$ of the induced Laplace-Beltrami operator on $\Sigma$ satisfies $\lambda_1 \geq \frac{n}{2}+ a_n(\Lambda^6 + b_n)^{-1}$, where $a_n$ and $b_n$ are explicit dimensional constants and $\Lambda$ is an upper bound for the length of the second fundamental form of $\Sigma$. This provides the first explicitly computable improvement on Choi & Wang's lower bound $\lambda_1 \geq \frac{n}{2}$ without any further assumptions on $\Sigma$.