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Conformal and extrinsic upper bounds for the harmonic mean of Neumann and Steklov eigenvalues

Hang Chen

TL;DR

The article derives sharp upper bounds for the harmonic mean of the first $m$ nonzero Neumann and Steklov eigenvalues on compact $m$-manifolds with boundary, expressing them through conformal volume $V_c(M,n)$ and relative conformal volume $V_{rc}(M,n)$. It establishes extrinsic Reilly-type bounds for closed submanifolds in space forms and provides general versions that apply to ambient manifolds admitting conformal immersions into spheres, using the generalized mean curvature $H_T$ and the operator $L_T$. Equality cases are carefully characterized, often forcing minimal or $T$-minimal immersions in spheres or geodesic spheres, with explicit geometric rigidity statements. The work unifies intrinsic spectral bounds with conformal geometry and extrinsic geometry, extending classical results for the first eigenvalue to the harmonic mean of the first $m$ eigenvalues and to Steklov problems, including surface-specific corollaries and higher-dimensional generalizations.

Abstract

Let $M$ be an $m$-dimensional compact Riemannian manifold with boundary. We obtain the upper bound of the harmonic mean of the first $m$ nonzero Neumann eigenvalues and Steklov eigenvalues involving the conformal volume and relative conformal volume, respectively. We also give an optimal sharp extrinsic upper bound for closed submanifolds in space forms. These extend the previous related results for the first nonzero eigenvalues.

Conformal and extrinsic upper bounds for the harmonic mean of Neumann and Steklov eigenvalues

TL;DR

The article derives sharp upper bounds for the harmonic mean of the first nonzero Neumann and Steklov eigenvalues on compact -manifolds with boundary, expressing them through conformal volume and relative conformal volume . It establishes extrinsic Reilly-type bounds for closed submanifolds in space forms and provides general versions that apply to ambient manifolds admitting conformal immersions into spheres, using the generalized mean curvature and the operator . Equality cases are carefully characterized, often forcing minimal or -minimal immersions in spheres or geodesic spheres, with explicit geometric rigidity statements. The work unifies intrinsic spectral bounds with conformal geometry and extrinsic geometry, extending classical results for the first eigenvalue to the harmonic mean of the first eigenvalues and to Steklov problems, including surface-specific corollaries and higher-dimensional generalizations.

Abstract

Let be an -dimensional compact Riemannian manifold with boundary. We obtain the upper bound of the harmonic mean of the first nonzero Neumann eigenvalues and Steklov eigenvalues involving the conformal volume and relative conformal volume, respectively. We also give an optimal sharp extrinsic upper bound for closed submanifolds in space forms. These extend the previous related results for the first nonzero eigenvalues.
Paper Structure (22 sections, 20 theorems, 129 equations)

This paper contains 22 sections, 20 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Conformal upper bounds for Neumann eigenvalues
  3. Conformal upper bounds for Steklov eigenvalues
  4. Reilly inequalities in space forms
  5. Extrinsic estimates for submanifolds of the Euclidean space
  6. Preliminaries
  7. Conformal relations: the ambient manifolds
  8. Conformal relations: the submanifolds
  9. The generalization of the mean curvature.
  10. Proofs for the Neumann eigenvalues
  11. The inequality in \ref{['eq-conf-h']}
  12. Equality in \ref{['eq-conf-h']}
  13. A corollary for submanifolds of a sphere
  14. Proofs for the Steklov eigenvalues
  15. The inequality in \ref{['eq-FS-h']}
  16. ...and 7 more sections

Key Result

Theorem 1.2

Let $(M,g)$ be an $m$-dimensional compact Riemannian manifold. Then the first nonzero eigenvalue $\lambda_1$ of the Laplacian (with Neumann boundary condition if $\partial M\neq \emptyset$) satisfies for all $n$ for which $V_{c}(M, n)$ is defined. Equality implies (C1). If $M$ is closed, then (C1) implies equality as well.

Theorems & Definitions (34)

  • Definition 1.1: cf. LY82
  • Theorem 1.2: LY82ESI86
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem 1.6: FS11*Theorem 6.2
  • Definition 1.7: cf. FS11
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 24 more