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 & Wang's lower bound $\lambda_1 \geq \frac{ n}{ 2}$ without any further assumptions on $\Sigma$.
Categories: math.DG math.AP math.SP
