An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere

Suppose that ${\mathrm{\Sigma }}^n\subset {\mathbb{S}}^{n+1}$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue ${\lambda }_1$ of the induced Laplace-Beltrami operator on $\mathrm{\Sigma }$ satisfies ${\lambda }_1\ge \frac{n}{2}+a_n({\mathrm{\Lambda }}^6+b_n{)}^{-1}$, where $a_n$ and $b_n$ are explicit dimensional constants and $\mathrm{\Lambda }$ is an upper bound for the length of the second fundamental form of $\mathrm{\Sigma }$. This provides the first explicitly computable improvement on Choi & Wang’s lower bound ${\lambda }_1\ge \frac{n}{2}$ without any further assumptions on $\mathrm{\Sigma }$.