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Large Steklov eigenvalues under volume constraints

Alexandre Girouard, Panagiotis Polymerakis

Abstract

In this note we establish an expression for the Steklov spectrum of warped products in terms of auxiliary Steklov problems for drift Laplacians with weight induced by the warping factor. As an application, we show that a compact manifold with connected boundary diffeomorphic to a product admits a family of Riemannian metrics which coincide on the boundary, have fixed volume and arbitrarily large first non-zero Steklov eigenvalue. These are the first examples of Riemannian metrics with these properties on three-dimensional manifolds.

Large Steklov eigenvalues under volume constraints

Abstract

In this note we establish an expression for the Steklov spectrum of warped products in terms of auxiliary Steklov problems for drift Laplacians with weight induced by the warping factor. As an application, we show that a compact manifold with connected boundary diffeomorphic to a product admits a family of Riemannian metrics which coincide on the boundary, have fixed volume and arbitrarily large first non-zero Steklov eigenvalue. These are the first examples of Riemannian metrics with these properties on three-dimensional manifolds.
Paper Structure (8 sections, 11 theorems, 78 equations)

This paper contains 8 sections, 11 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. The Steklov spectrum of warped products
  3. Plan
  4. Preliminary material
  5. Mixed Steklov--Neumann spectrum of diffusion and Laplace type operators
  6. Quasi-isometries and eigenvalues
  7. The Steklov spectrum of warped products
  8. Large eigenvalues on product manifolds

Key Result

Theorem 2

Let $(M,g_M)$ be the Riemannian product of a compact manifold $B^n$ with boundary and a closed manifold $F^k$, where $n > k\geq 1$. Then for any $\varepsilon > 0$ there exists a Riemannian metric $g_\varepsilon$ on $M$ that coincides with $g_M$ in a neighborhood of $\partial M$, and satisfies $dV_{g

Theorems & Definitions (18)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Remark 8
  • Proposition 9
  • Theorem 10
  • ...and 8 more