The first eigenvalue of embedded minimal hypersurfaces in the unit sphere

TL;DR

This paper establishes a new lower bound for the first eigenvalue of embedded minimal hypersurfaces in spheres, improving previous results and providing uniform curvature estimates when the second fundamental form is constant.

Contribution

It introduces a sharper lower bound for the first eigenvalue, generalizes prior work, and offers uniform scalar curvature estimates for hypersurfaces with constant second fundamental form.

Findings

Lower bound for  exceeds (m/2) by a positive constant.

Improved eigenvalue estimate without proving Chern's conjecture.

Provides a uniform scalar curvature estimate for hypersurfaces with constant .

Abstract

In this article, we prove that for an embedded minimal hypersurface $\Sigma^{m}$ in $S^{m+1}$, the first eigenvalue $\lambda_1$ of the Laplacian operator on $\Sigma$ satisfies: $$\lambda_1> \frac{m}{2}+G(m, |A|_{\max}, |A|_{\min} ) ,$$ where $|A|_{\max}$ and $|A|_{\min}$ denote the maximum and minimum of the norm of the second fundamental form on $\Sigma$, respectively; $G(m, |A|_{\max}, |A|_{\min} )$ is a positive constant that depends only on $m,|A|_{\max}, |A|_{\min}$. In particular, when the norm $|A|$ of the second fundamental form is constant, we can obtain a gap depending only on $m$, i.e., $$\lambda_1>\left(\frac{1}{2}+ c \right)m ,$$ where $c$ is a positive absolute constant. This improves Choi and Wang's previous result \cite{chw1983first} that $\lambda_1\geq \frac{m}{2}$. Our result shows that one can improve Choi and Wang's result directly without proving Chern's conjecture. This also generalizes Tang and Yan's work \cite{tangyan2013isoparametric}. Based on the proof of the result above, using the lower bound of the first Steklov eigenvalue, we prove that if the norm $|A|$ of the second fundamental form is constant, then $$|A| \leq \frac{C(m)\textup{Volume}(\Sigma)}{\textup{Volume}(S^m)},$$ where $C(m)$ is a constant that depends only on $m$. This provides a uniform estimate for the scalar curvature of embedded minimal hypersurfaces with constant norm of the second fundamental form. Moreover, this may be useful for Chern's problem.