An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere

Author:

Jonah A. J. Duncan, Yannick Sire, Joel Spruck

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP), Spectral Theory (math.SP)

journal:

date:

2023-08-22 16:00:00

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 $\lambda_1 \geq \frac{n}{2}+ a_n(\Lambda^6 + b_n)^{-1}$, 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 $\Sigma$.

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