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Gromov-Hausdorff convergence of metric pairs and metric tuples

Andrés Ahumada Gómez, Mauricio Che

Abstract

We study the Gromov-Hausdorff convergence of metric pairs and metric tuples and prove the equivalence of different natural definitions of this concept. We also prove embedding, completeness and compactness theorems in this setting. Finally, we get a relative version of Fukaya's theorem about quotient spaces under Gromov--Hausdorff equivariant convergence and a version of Grove-Petersen--Wu's finiteness theorem for stratified spaces.

Gromov-Hausdorff convergence of metric pairs and metric tuples

Abstract

We study the Gromov-Hausdorff convergence of metric pairs and metric tuples and prove the equivalence of different natural definitions of this concept. We also prove embedding, completeness and compactness theorems in this setting. Finally, we get a relative version of Fukaya's theorem about quotient spaces under Gromov--Hausdorff equivariant convergence and a version of Grove-Petersen--Wu's finiteness theorem for stratified spaces.
Paper Structure (8 sections, 29 theorems, 123 equations)

This paper contains 8 sections, 29 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Gromov--Hausdorff convergence of metric pairs
  3. Compact case
  4. Proper length spaces
  5. Non-compact case
  6. Main results
  7. Convergence of tuples
  8. Applications

Key Result

Theorem 1.1

Let $\{(X_i,A_i)\}_{i\in\mathbb{N}}$ be a sequence of proper metric pairs. Suppose that Then there exist a non-complete locally complete metric space $Y$ and a closed subset $W\subset \overline{Y} \setminus Y=:Z$, where $\overline{Y}$ is the metric completion of $Y$, with the following properties:

Theorems & Definitions (72)

  • Theorem 1.1: Embedding theorem
  • Theorem 1.2: Completeness theorem
  • Theorem 1.3: Compactness theorem
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 62 more