An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere

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 ${\lambda }_1\ge \frac{n}{2}+a_n({\Lambda }^6+b_n{)}^{-1}$, 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 ${\lambda }_1\ge \frac{n}{2}$ without any further assumptions on $\Sigma$.