Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some rigidity theorems for spectral curvature bounds

Xiaoxiang Chai, Yukai Sun

Abstract

We investigate the geometric implications of spectral curvature bounds, extending classical rigidity results in scalar curvature geometry to the spectral setting. By systematically employing the warped $μ$-bubble method, we show classification theorems for stable weighted minimal hypersurfaces in 3-manifolds with nonnegative spectral scalar curvature, and we establish band width estimates for both spectral Ricci and spectral scalar curvatures. Furthermore, we prove some splitting theorems under spectral curvature conditions, including a spectral version of the Geroch conjecture for manifolds with arbitrary ends and a result related to the Milnor conjecture.

Some rigidity theorems for spectral curvature bounds

Abstract

We investigate the geometric implications of spectral curvature bounds, extending classical rigidity results in scalar curvature geometry to the spectral setting. By systematically employing the warped -bubble method, we show classification theorems for stable weighted minimal hypersurfaces in 3-manifolds with nonnegative spectral scalar curvature, and we establish band width estimates for both spectral Ricci and spectral scalar curvatures. Furthermore, we prove some splitting theorems under spectral curvature conditions, including a spectral version of the Geroch conjecture for manifolds with arbitrary ends and a result related to the Milnor conjecture.

Paper Structure

This paper contains 24 sections, 25 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Weighted minimal hypersurfaces
  3. Constant weighted mean curvature
  4. Band width estimates
  5. Applications and extensions
  6. Basics of warped $\mu$-bubbles
  7. Warped $\mu$-bubble in bands
  8. Variation of a warped $h$-hypersurface
  9. Rewrite of second variation
  10. Classification of stable weighted minimal surfaces
  11. Classification of stable surfaces
  12. Hypersurfaces of constant weighted mean curvature
  13. Spectral scalar curvature case
  14. Spectral Ricci curvature case
  15. Band width estimates with spectral curvature bounds
  16. ...and 9 more sections

Key Result

Theorem 1.4

Let $0 \leqslant \gamma <4$, $(M^3, g)$ be a 3-dimensional complete manifold with spectral nonnegative scalar curvature and $\Sigma$ be a stable, complete, oriented weighted minimal surface with weight $u^{\gamma}$ in $(M, g)$. Then there are two possibilities: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (58)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 48 more