A Proof of the Eigenvalue Ratio Bound for Embedded Surfaces

Abstract

We explain how the spectrum of a closed embedded surface $\hat{I}\pounds \subset {\mathbb{R}}^3$ relates to the Dirichlet spectrum of the bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$ with $\partial \hat{I}\text{c◯}=\hat{I}\pounds$. We prove that there exists a positive constant $K_g$, depending only on the genus $g$ of $\hat{I}\pounds$, such that $\hat{I}{»}_k^D(\hat{I}\text{c◯}{)}^{3/2}/(\hat{I}{»}_k(\hat{I}\pounds )\sqrt{\hat{I}{»}_1(\hat{I}\pounds )})\ge K_g$, where $\hat{I}{»}_k(\hat{I}\pounds )$ denotes the $k$-th nonzero eigenvalue of the Laplace-Beltrami operator on $\hat{I}\pounds$ and $\hat{I}{»}_k^D(\hat{I}\text{c◯})$ denotes the $k$-th eigenvalue of the Laplacian on $\hat{I}\text{c◯}$ with Dirichlet boundary conditions. Moreover, we explicitly obtain the dependence of $K_g$ on the genus, showing that $K_g\propto (g+1{)}^{-1}$, and we determine the optimal constant $K_0$ for $k=1$ in the genus-zero case. A generalized version of this result in arbitrary dimension is also provided for domains whose boundaries have nonnegative Ricci curvature.