Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Llarull type theorems for bands in Three and Four dimensions

Xiaoxiang Chai, Xueyuan Wan

Abstract

Llarull's theorem asserts that the scalar curvature and the metric on the $n$-sphere cannot be bounded below at the same time by those of the standard $n$-sphere. Using the warped $μ$-bubble method, we develop Llarull type theorems for three and four-dimensional bands with spectral scalar curvature bounds.

Llarull type theorems for bands in Three and Four dimensions

Abstract

Llarull's theorem asserts that the scalar curvature and the metric on the -sphere cannot be bounded below at the same time by those of the standard -sphere. Using the warped -bubble method, we develop Llarull type theorems for three and four-dimensional bands with spectral scalar curvature bounds.
Paper Structure (12 sections, 20 theorems, 95 equations)

This paper contains 12 sections, 20 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Basics of warped $\mu$-bubbles
  3. The variational problem
  4. Rewrite of the second variation
  5. Model metrics for Theorem \ref{['spec llarull band']}
  6. Llarull type theorem for bands
  7. Construction of $\mu$
  8. Local rigidity
  9. Analysis of the foliation
  10. Llarull's theorem with conical end points
  11. Tangent cone analysis
  12. Llarull's theorem

Key Result

Theorem 1.1

Let $f : (M, g) \to (\mathbb{S}^n, g_{\mathbb{S}^n})$ be a distance non-increasing spin map such that $R_g \geqslant n (n - 1)$, then $f$ is an isometry.

Theorems & Definitions (42)

  • Theorem 1.1: llarull-sharp-1998
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: listing-scalar-arxiv-2010
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 32 more