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Band width estimates with lower spectral curvature bounds

Xiaoxiang Chai, Yukai Sun

TL;DR

The paper develops upper bounds for the width of torical bands under lower spectral curvature bounds using warped $\mu$-bubble methods. It treats two cases—spectral Ricci and spectral scalar curvature bounds—proving explicit width bounds and achieving rigidity characterizations that identify model torical bands realizing equality. The approach combines variational formulas for warped $\mu$-bubbles, existence and stability results, and a foliation-based rigidity argument to conclude that the ambient metric must assume a (doubly) warped product form in the rigidity regime. These results extend prior qualitative width estimates by providing quantitative bounds and explicit model metrics in dimensions $3\le n\le 7$. The work thereby connects spectral curvature bounds to geometric constraints on band width and reinforces the warped $\mu$-bubble framework as a robust tool for curvature-driven rigidity phenomena.

Abstract

In this work, we use the warped \( μ\)-bubble method to study the consequences of a spectral curvature bound. In particular, with a lower spectral Ricci curvature bound and lower spectral scalar curvature bound, we show that the band width of a torical band is bounded above. We also obtain some rigidity results.

Band width estimates with lower spectral curvature bounds

TL;DR

The paper develops upper bounds for the width of torical bands under lower spectral curvature bounds using warped -bubble methods. It treats two cases—spectral Ricci and spectral scalar curvature bounds—proving explicit width bounds and achieving rigidity characterizations that identify model torical bands realizing equality. The approach combines variational formulas for warped -bubbles, existence and stability results, and a foliation-based rigidity argument to conclude that the ambient metric must assume a (doubly) warped product form in the rigidity regime. These results extend prior qualitative width estimates by providing quantitative bounds and explicit model metrics in dimensions . The work thereby connects spectral curvature bounds to geometric constraints on band width and reinforces the warped -bubble framework as a robust tool for curvature-driven rigidity phenomena.

Abstract

In this work, we use the warped -bubble method to study the consequences of a spectral curvature bound. In particular, with a lower spectral Ricci curvature bound and lower spectral scalar curvature bound, we show that the band width of a torical band is bounded above. We also obtain some rigidity results.

Paper Structure

This paper contains 15 sections, 21 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Basics of warped mu-bubble
  3. First and second variations of warped mu-bubble
  4. Existence theorem
  5. Band width estimate with the spectral Ricci curvature condition
  6. Proof of the band width estimate
  7. Existence of a non-trivial minimiser
  8. Construction of foliation
  9. Rigidity
  10. Band width estimate with the spectral scalar curvature condition
  11. Proof of the band width estimate
  12. Existence of a non-trivial minimiser
  13. Rigidity
  14. Determine the metric in the spectral Ricci case
  15. Band width estimate with Ricci curvature bound

Key Result

Theorem 1.1

For a Riemannian band $M^{3}=[-1,1]\times T^{2}$ with the metric $g$, let $u$ be a positive smooth function on $M\setminus \partial M$ with $u=0$ on $\partial M$ such that where $\Lambda>0$, $0< \gamma<4$. Then

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: First variation of warped $\mu$-bubbles
  • Lemma 2.2: Second variation of warped $\mu$-bubbles
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5: Existence of $\mu$-bubble
  • ...and 32 more