Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral ratios and gaps for Steklov eigenvalues of balls with revolution-type metrics

Jade Brisson, Bruno Colbois, Katie Gittins

TL;DR

This work analyzes the Steklov spectrum on balls with revolution-type metrics, aiming to bound ratios and gaps of the eigenvalues without curvature or boundary convexity assumptions. By reducing to radial ODEs for the angular modes and employing Rayleigh-quotient comparisons, the authors establish sharp upper bounds for spectral ratios in dimensions $n\ge 3$ and, in 3D, for spectral gaps; they also obtain gap bounds in higher dimensions under mild metric constraints. A key achievement is showing that the ratio bounds are optimal in the sense that the supremum over admissible metrics equals the corresponding ratio of Laplacian eigenvalues on the sphere, with explicit constructions approaching this supremum. The results illuminate how geometry of the metric in revolution form controls the Steklov spectrum and provide precise, dimension-dependent extremal bounds. Methods combine spectral decomposition on $\mathbb{S}^{n-1}$, reduced radial ODEs, and delicate perturbation arguments to relate near-extremality to rigidity of radial profiles.

Abstract

We investigate upper bounds for the spectral ratios and gaps for the Steklov eigenvalues of balls with revolution-type metrics. We do not impose conditions on the Ricci curvature or on the convexity of the boundary. We obtain optimal upper bounds for the Steklov spectral ratios in dimensions 3 and higher. In dimension 3, we also obtain optimal upper bounds for the Steklov spectral gaps. By imposing additional constraints on the metric, we obtain upper bounds for the Steklov spectral gaps in dimensions 4 and higher.

Spectral ratios and gaps for Steklov eigenvalues of balls with revolution-type metrics

TL;DR

This work analyzes the Steklov spectrum on balls with revolution-type metrics, aiming to bound ratios and gaps of the eigenvalues without curvature or boundary convexity assumptions. By reducing to radial ODEs for the angular modes and employing Rayleigh-quotient comparisons, the authors establish sharp upper bounds for spectral ratios in dimensions and, in 3D, for spectral gaps; they also obtain gap bounds in higher dimensions under mild metric constraints. A key achievement is showing that the ratio bounds are optimal in the sense that the supremum over admissible metrics equals the corresponding ratio of Laplacian eigenvalues on the sphere, with explicit constructions approaching this supremum. The results illuminate how geometry of the metric in revolution form controls the Steklov spectrum and provide precise, dimension-dependent extremal bounds. Methods combine spectral decomposition on , reduced radial ODEs, and delicate perturbation arguments to relate near-extremality to rigidity of radial profiles.

Abstract

We investigate upper bounds for the spectral ratios and gaps for the Steklov eigenvalues of balls with revolution-type metrics. We do not impose conditions on the Ricci curvature or on the convexity of the boundary. We obtain optimal upper bounds for the Steklov spectral ratios in dimensions 3 and higher. In dimension 3, we also obtain optimal upper bounds for the Steklov spectral gaps. By imposing additional constraints on the metric, we obtain upper bounds for the Steklov spectral gaps in dimensions 4 and higher.
Paper Structure (8 sections, 9 theorems, 110 equations)

This paper contains 8 sections, 9 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Tools in this geometric setting
  3. Proofs of main results
  4. Proof of Theorem \ref{['thm1']}
  5. Proof of Theorem \ref{['thm2']}
  6. Proof of Theorem \ref{['thm3']}
  7. Proof of Theorem \ref{['thm:revolution']}
  8. Upper bounds for Steklov spectral gaps

Key Result

Theorem 1.1

Let $M= [0,R] \times \mathbb{S}^{n-1}$ be equipped with the metric $g_h = dr^2 + h(r)^2 g_0,$ where $g_0$ is the usual metric on the $(n-1)$-dimensional unit sphere and $h$ satisfies assumptions $(H)$. For $k \geq 1$ we have that

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm1']}
  • ...and 12 more