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Cheeger-Gromov compactness for manifolds with boundary

Olaf Müller

Abstract

We prove Cheeger-Gromov convergence for a subsequence of a given sequence of manifolds-with-boundary of bounded geometry. The method of the proof is to reduce, via height functions, the problem to the setting of Hamilton's compactnes theorem for manifolds without boundary.

Cheeger-Gromov compactness for manifolds with boundary

Abstract

We prove Cheeger-Gromov convergence for a subsequence of a given sequence of manifolds-with-boundary of bounded geometry. The method of the proof is to reduce, via height functions, the problem to the setting of Hamilton's compactnes theorem for manifolds without boundary.

Paper Structure

This paper contains 16 sections, 6 theorems, 21 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. Motivation
  3. Bounded geometry and main result
  4. Acknowledgments.
  5. Notation, Convergence for manifolds with boundary
  6. Conventions, notation, definition of convergence
  7. Gromov-Hausdorff convergence
  8. Smooth Cheeger-Gromov convergence
  9. Proof of the main result (Theorem A)
  10. Step 1: An extension procedure
  11. Height functions
  12. Extending functions beyond a boundary
  13. Proof of Theorem \ref{['MetricExtension']}
  14. Step 2: Convergence with height functions
  15. Step 3: Coming back to manifolds-with-boundary
  16. ...and 1 more sections

Key Result

Theorem 2.2

Let $A: i \mapsto (X_i,g_i,x^0_i)$ be a sequence of pointed complete $n$-dimensional Riemannian manifolds such that $\mathop{\mathrm{{\text{\rm Ric}}}}\nolimits_{g_i}\geq (n-1)\kappa$ for some $\kappa\in \mathbb{R}$ and all $i \in \mathbb{N}$. Then there is a pointed proper complete metric space $(Y

Theorems & Definitions (17)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3: from Bam, Sec. 3.2.1
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 3.1
  • ...and 7 more