A Proof of the Eigenvalue Ratio Bound for Embedded Surfaces

We explain how the spectrum of a closed embedded surface $\Sigma \subset \mathbb{R}^3$ relates to the Dirichlet spectrum of the bounded domain $\Omega \subset \mathbb{R}^3$ with $\partial \Omega = \Sigma$. We prove that there exists a positive constant $K_g$, depending only on the genus $g$ of $\Sigma$, such that $\lambda_k^D(\Omega)^{3/2}/(\lambda_k(\Sigma)\sqrt{\lambda_1(\Sigma)}) \ge K_g$, where $\lambda_k(\Sigma)$ denotes the $k$-th nonzero eigenvalue of the Laplace-Beltrami operator on $\Sigma$ and $\lambda_k^D(\Omega)$ denotes the $k$-th eigenvalue of the Laplacian on $\Omega$ 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.