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Filling Radius, Quantitative $K$-theory and Positive Scalar Curvature

Jinmin Wang, Zhizhang Xie, Guoliang Yu, Bo Zhu

Abstract

We prove a quantitative upper bound on the filling radius of complete, spin manifolds with uniformly positive scalar curvature using the quantitative operator $K$-theory and index theory.

Filling Radius, Quantitative $K$-theory and Positive Scalar Curvature

Abstract

We prove a quantitative upper bound on the filling radius of complete, spin manifolds with uniformly positive scalar curvature using the quantitative operator -theory and index theory.
Paper Structure (12 sections, 18 theorems, 145 equations)

This paper contains 12 sections, 18 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. A basic introduction to Filling Radius
  3. Kuratowski embedding
  4. Filling radius
  5. Filling radius and coarse geometry
  6. Lipschitz properties of spaces with finite asymptotic dimension
  7. Proof of Theorem \ref{['thm:main']}
  8. Index pairing
  9. Proof of the main theorem
  10. Metric invariants of Riemannian manifolds
  11. Metric invariants
  12. Metric invariants and uniformly positive scalar curvature

Key Result

Theorem 1.2

Let $(M^n,g)$ be a complete, noncompact, spin Riemannian manifold with finite asymptotic dimension $m$ with control function $\mathcal{D}$. If $(M,g)$ has uniformly positive scalar curvature $Sc_g\geq \sigma^2 >0$ and bounded geometry, then there exists a constant $c=c(\sigma, m, \mathcal{D})$ such

Theorems & Definitions (55)

  • Conjecture 1.1: gromovmetricstructure
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2: Gromov gromovfilling
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Example 2.6
  • ...and 45 more