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The first Steklov eigenvalue on manifolds with nonnegative Ricci curvature and convex boundary

Jonah A. J. Duncan, Aditya Kumar

Abstract

We establish a new lower bound for the first non-zero Steklov eigenvalue of a compact Riemannian manifold with non-negative Ricci curvature and (strictly) convex boundary. Related results are also obtained under weaker geometric hypotheses.

The first Steklov eigenvalue on manifolds with nonnegative Ricci curvature and convex boundary

Abstract

We establish a new lower bound for the first non-zero Steklov eigenvalue of a compact Riemannian manifold with non-negative Ricci curvature and (strictly) convex boundary. Related results are also obtained under weaker geometric hypotheses.

Paper Structure

This paper contains 9 sections, 13 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Lower bounds on the first eigenvalue and Escobar's conjecture
  3. Overview of results
  4. An initial lemma and summary of proofs
  5. Proofs of main results
  6. Proof of Theorem \ref{['A']} and Corollary \ref{['B']}
  7. Proof of Theorem \ref{['E']} and Theorem \ref{['F']}
  8. Proof of Theorem \ref{['C']}
  9. A spectral gap result when $\operatorname{Ric}_g<0$

Key Result

Theorem A

Let $(M^2,g)$ be a smooth compact surface with non-empty boundary $\Sigma$. Let $K_{g}$ denote the Gaussian curvature of $(M,g)$ and $\kappa_{g}$ the geodesic curvature of $\Sigma$. Then and equality holds if and only if $M$ is a Euclidean ball with radius $\frac{1}{\kappa}$.

Theorems & Definitions (33)

  • Theorem A: E97
  • Theorem B: E97
  • Conjecture : E99
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • ...and 23 more