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A Proof of the Eigenvalue Ratio Bound for Embedded Surfaces

Ricardo Gloria-Picazzo, Yingying Wu, Shing-Tung Yau

Abstract

We explain how the spectrum of a closed embedded surface $Σ\subset \mathbb{R}^3$ relates to the Dirichlet spectrum of the bounded domain $Ω\subset \mathbb{R}^3$ with $\partial Ω= Σ$. We prove that there exists a positive constant $K_g$, depending only on the genus $g$ of $Σ$, such that $λ_k^D(Ω)^{3/2}/(λ_k(Σ)\sqrt{λ_1(Σ)}) \ge K_g$, where $λ_k(Σ)$ denotes the $k$-th nonzero eigenvalue of the Laplace-Beltrami operator on $Σ$ and $λ_k^D(Ω)$ denotes the $k$-th eigenvalue of the Laplacian on $Ω$ 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.

A Proof of the Eigenvalue Ratio Bound for Embedded Surfaces

Abstract

We explain how the spectrum of a closed embedded surface relates to the Dirichlet spectrum of the bounded domain with . We prove that there exists a positive constant , depending only on the genus of , such that , where denotes the -th nonzero eigenvalue of the Laplace-Beltrami operator on and denotes the -th eigenvalue of the Laplacian on with Dirichlet boundary conditions. Moreover, we explicitly obtain the dependence of on the genus, showing that , and we determine the optimal constant for 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.
Paper Structure (9 sections, 8 theorems, 51 equations)

This paper contains 9 sections, 8 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Eigenvalues on Riemannian Manifolds
  3. Spectral bounds for the Laplacian
  4. Upper bound for the volume of $\Omega$ in terms of $\lambda_1(\Sigma)$ and $\mathop{\mathrm{area}}\nolimits(\Sigma)$
  5. Berezin--Li--Yau inequality
  6. Upper bounds for $\lambda_k(\Sigma)$ in terms of the genus
  7. Proof on the eigenvalue ratio for embedded surfaces
  8. On the generalization to any dimension
  9. Proof of Proposition \ref{['prop:bound']}

Key Result

Theorem 1.1

If $\Sigma\subset\mathbb{R}^{3}$ is an embedded closed connected surface of genus $g$ equipped with the induced Euclidean metric and let $\Omega\subset\mathbb{R}^{3}$ be the bounded domain enclosed by $\Sigma$, then where $C$ is an absolute positive constant and $C_3^{3/2}=6\pi^{2}$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: reilly1977first
  • Theorem 2.2: Berezin--Li--Yau; Li--Yau li1983schrodinger; Berezin berezin1972covariant
  • Corollary 2.3: Li--Yau bound li1983schrodinger
  • proof
  • Theorem 2.4: Theorem 0.5 korevaar1993upper - see also Theorem 5.4 in grigor2004eigenvalues
  • Remark 2.5
  • proof : Proof of \ref{['thm:main_res']}
  • Proposition 3.1: Genus $0$, $k=1$: optimal constant and minimizer
  • ...and 5 more