The first eigenvalue of embedded minimal hypersurfaces in the unit sphere

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.