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A generalization of the Perelman gluing theorem and applications

Philipp Reiser, David J. Wraith

Abstract

We extend a positive Ricci curvature gluing theorem of Perelman to a range of positive intermediate curvature conditions, ranging from positive scalar curvature up to (and including) positive sectional curvature. As an application of this, we demonstrate that the observer moduli space of metrics with positive intermediate Ricci curvatures can have non-trivial higher homotopy groups. Further applications include deriving a sufficient condition for the existence of a metric with positive intermediate Ricci curvature and totally geodesic boundary.

A generalization of the Perelman gluing theorem and applications

Abstract

We extend a positive Ricci curvature gluing theorem of Perelman to a range of positive intermediate curvature conditions, ranging from positive scalar curvature up to (and including) positive sectional curvature. As an application of this, we demonstrate that the observer moduli space of metrics with positive intermediate Ricci curvatures can have non-trivial higher homotopy groups. Further applications include deriving a sufficient condition for the existence of a metric with positive intermediate Ricci curvature and totally geodesic boundary.
Paper Structure (7 sections, 17 theorems, 36 equations)

This paper contains 7 sections, 17 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. The $C^1$-smoothing
  4. The $C^\infty$-smoothing
  5. Proof of Corollary \ref{['almost-nonnegative']}
  6. Proof of Corollaries \ref{['tot_geodesic']} and \ref{['double']}
  7. Proof of Theorem \ref{['obs']}

Key Result

Theorem 1

Let $(M_1, h_1)$ and $(M_2, h_2)$ be a pair of $n$-dimensional Riemannian manifolds and let $\phi:(\partial M_1, h_1|_{\partial M_1})\rightarrow (\partial M_2, h_2|_{\partial M_2} )$ be an isometry of the boundaries, which we will assume to be compact. Moreover, the smoothed metric can be chosen to agree with the original metrics outside an arbitrarily small neighbourhood of the gluing area.

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Lemma 2.1: cf. also Bu3 and BWW
  • proof
  • Lemma 2.2
  • ...and 24 more