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Estimates for low Steklov eigenvalues of surfaces with several boundary components

Hélène Perrin

Abstract

In this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $σ_1$ of a compact connected 2-dimensional Riemannian manifold $M$ with several cylindrical boundary components. These estimates show how the geometry of $M$ away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

Estimates for low Steklov eigenvalues of surfaces with several boundary components

Abstract

In this article, we give computable lower bounds for the first non-zero Steklov eigenvalue of a compact connected 2-dimensional Riemannian manifold with several cylindrical boundary components. These estimates show how the geometry of away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.
Paper Structure (13 sections, 13 theorems, 57 equations, 4 figures)

This paper contains 13 sections, 13 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Cheeger-type estimates and mixed problems
  3. Steklov eigenvalues
  4. Cheeger-type estimates
  5. Explicit estimates for Steklov eigenvalues
  6. Lower bounds for $\sigma_1$ of surfaces with several cylindrical boundary components
  7. Geometric bounds on the low Steklov eigenvalues of a compact hyperbolic surface with geodesic boundary
  8. Step 1: $\mathop{\mathrm{l}}\nolimits_n\leq\beta_1$ where $\beta_1$ is a constant depending only on $g$ and $b$.
  9. Step 2: $\sigma_n\leq C_2 \frac{\mathop{\mathrm{l}}\nolimits_n}{a}$.
  10. Step 3: $C_1\mathop{\mathrm{l}}\nolimits_n^2\leq \sigma_n$.
  11. Estimation of $h_1(\Omega_i)$
  12. Estimation of $h_2(\Omega_i)$
  13. Acknowledgements

Key Result

Theorem 1

Let $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold with a boundary having $b\geq 2$ components of length $a$. Assume that the boundary $\partial M=\Sigma_1\cup\dots\cup\Sigma_b$ has a neighborhood $V(\partial M)$ which is isometric to the union of disjoint right cylinders $\cup_{

Figures (4)

  • Figure 1: A cylinder on which we have glued a surface of revolution
  • Figure 2: A surface with two cylindrical boundary neighborhoods connected by a thin cylinder
  • Figure 3: The domain $A=\cup_{i=1}^{b}(\Sigma_i\times [0,\mathop{\mathrm{inj}}\nolimits_{\partial M}(M)))\cup(\cup_{i=2}^{b}T_i)$
  • Figure 4: A schematic representation of possible configurations of $D$ in $\Omega_i$ corresponding to each of the five cases

Theorems & Definitions (35)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Definition 3
  • Theorem 3
  • Remark 1
  • Definition 4
  • Remark 2
  • Remark 3
  • ...and 25 more