An Improved Eigenvalue Estimate for Embedded Minimal Hypersurfaces in the Sphere

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 and Wang’s lower bound $\lambda _{1} \geq \frac{n}{2}$ without any further assumptions on $\Sigma $.