Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scaling inequalities for Steklov eigenvalues in space forms and sharp eigenvalue estimates on warped product manifolds

Zongyi Lv, Changwei Xiong, Yuxun Zou

TL;DR

This work develops a comprehensive framework for scaling and monotonicity of Steklov eigenvalues on space forms and warped product manifolds. By exploiting four natural geometric factors and separable solutions on geodesic balls, it provides explicit 2D formulas, higher-dimensional monotonicity results, and sharp bounds for fourth-order Steklov problems under nonnegative Ricci curvature and convex boundaries. A central achievement is the verification of Wang–Xia’s conjecture in dimension three within warped-product geometry, along with Escobar-type and unimodal curvature-dependent bounds. The results advance sharp eigenvalue estimates on curved spaces and contribute tools for comparison-type spectral geometry in manifolds with boundary.

Abstract

In the first part, we derive monotonicity of the normalized spectra for the second-order Steklov problem and two fourth-order Steklov problems on the $2$-dimensional geodesic disks with respect to the geodesic radius in the sphere and the hyperbolic space. The normalizations are made using four natural geometric factors. As corollaries, we get Escobar-type bounds for Steklov eigenvalues on $2$-dimensional geodesic disks with varying curvature in space forms. We also get two monotonicity results for higher-dimensional cases. In the second part, we obtain some sharp bounds concerning the spectra of the two fourth-order Steklov problems on warped product manifolds with non-negative Ricci curvature and a strictly convex boundary. In particular, we confirm Qiaoling Wang and Changyu Xia's conjecture (2018) on the sharp lower bound of the first non-zero eigenvalue of a fourth-order Steklov problem in the case of $3$-dimensional warped product manifolds.

Scaling inequalities for Steklov eigenvalues in space forms and sharp eigenvalue estimates on warped product manifolds

TL;DR

This work develops a comprehensive framework for scaling and monotonicity of Steklov eigenvalues on space forms and warped product manifolds. By exploiting four natural geometric factors and separable solutions on geodesic balls, it provides explicit 2D formulas, higher-dimensional monotonicity results, and sharp bounds for fourth-order Steklov problems under nonnegative Ricci curvature and convex boundaries. A central achievement is the verification of Wang–Xia’s conjecture in dimension three within warped-product geometry, along with Escobar-type and unimodal curvature-dependent bounds. The results advance sharp eigenvalue estimates on curved spaces and contribute tools for comparison-type spectral geometry in manifolds with boundary.

Abstract

In the first part, we derive monotonicity of the normalized spectra for the second-order Steklov problem and two fourth-order Steklov problems on the -dimensional geodesic disks with respect to the geodesic radius in the sphere and the hyperbolic space. The normalizations are made using four natural geometric factors. As corollaries, we get Escobar-type bounds for Steklov eigenvalues on -dimensional geodesic disks with varying curvature in space forms. We also get two monotonicity results for higher-dimensional cases. In the second part, we obtain some sharp bounds concerning the spectra of the two fourth-order Steklov problems on warped product manifolds with non-negative Ricci curvature and a strictly convex boundary. In particular, we confirm Qiaoling Wang and Changyu Xia's conjecture (2018) on the sharp lower bound of the first non-zero eigenvalue of a fourth-order Steklov problem in the case of -dimensional warped product manifolds.
Paper Structure (16 sections, 18 theorems, 293 equations, 2 figures)

This paper contains 16 sections, 18 theorems, 293 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scaling inequalities in the $2$-dimensional sphere
  3. Scaling inequalities in the $2$-dimensional hyperbolic space
  4. Monotonicity of Steklov eigenvalues on geodesic disks with varying curvature
  5. Scaling inequalities for $n\geq 3$
  6. Sharp bounds for eigenvalues on warped product manifolds
  7. Preliminaries
  8. The general setting of warped product manifolds
  9. Geodesic balls and geometric factors in the sphere and the hyperbolic space
  10. Second-order and fourth-order Steklov eigenvalue problems
  11. Proofs of results in the $2$-dimensional sphere
  12. Proofs of results in the $2$-dimensional hyperbolic space
  13. Proofs of monotonicity of Steklov eigenvalues on geodesic disks with varying curvature
  14. Proofs of scaling inequalities for $n\geq 3$
  15. Proofs of sharp bounds for eigenvalues on warped product manifolds
  16. ...and 1 more sections

Key Result

Theorem 3

For the eigenvalue $\xi_{(m)}$$(m \geq 1)$ of the geodesic disk of radius $R$ in the $2$-dimensional sphere, we have the following conclusions.

Figures (2)

  • Figure 1: Geometric factors in $\mathbb S^2$
  • Figure 2: Geometric factors in $\mathbb H^2$

Theorems & Definitions (48)

  • Remark 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Theorem 7
  • Theorem 8
  • Remark 9
  • Corollary 10
  • ...and 38 more