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Gromov's Compactness Theorem for the Intrinsic Timed-Hausdorff Distance

M Che, R Perales, C Sormani

TL;DR

The paper addresses the problem of establishing a compactness theory for the intrinsic timed-Hausdorff convergence of timed-metric-spaces, enabling limits that preserve both metric and causal structure. It develops timed-Fréchet maps and introduces an address framework to mimic and strengthen Gromov's classical compactness, proving a Timed Gromov Compactness Theorem with uniform convergence controlled by an index set of addresses. A key contribution is the construction of a limit via a dense, uniformly indexed point family and the demonstration of convergence in the timed-Hausdorff sense using timed-Fréchet embeddings into a compact ambient space, along with an Arzelà–Ascoli type result in this setting. The work also derives GH-compactness and Arzelà–Ascoli results with addresses and proposes a conjecture for a stronger Arzelà–Ascoli theorem, highlighting potential applications to Lorentzian geometry and space-time convergence analyses.

Abstract

The intrinsic timed-Hausdorff distance between timed-metric-spaces, first introduced by Sakovich--Sormani, yields a weak notion of convergence for space-times. In this paper we prove a compactness theorem for the intrinsic timed-Hausdorff convergence of timed-metric-spaces using timed-Fréchet maps. Our proof introduces the notion of ``addresses'' and provides a new way of stating Gromov's original compactness theorem for Gromov--Hausdorff convergence of metric spaces. We also obtain a new Arzelà--Ascoli theorem for real valued uniformly bounded Lipschitz functions on Gromov--Hausdorff converging compact metric spaces.

Gromov's Compactness Theorem for the Intrinsic Timed-Hausdorff Distance

TL;DR

The paper addresses the problem of establishing a compactness theory for the intrinsic timed-Hausdorff convergence of timed-metric-spaces, enabling limits that preserve both metric and causal structure. It develops timed-Fréchet maps and introduces an address framework to mimic and strengthen Gromov's classical compactness, proving a Timed Gromov Compactness Theorem with uniform convergence controlled by an index set of addresses. A key contribution is the construction of a limit via a dense, uniformly indexed point family and the demonstration of convergence in the timed-Hausdorff sense using timed-Fréchet embeddings into a compact ambient space, along with an Arzelà–Ascoli type result in this setting. The work also derives GH-compactness and Arzelà–Ascoli results with addresses and proposes a conjecture for a stronger Arzelà–Ascoli theorem, highlighting potential applications to Lorentzian geometry and space-time convergence analyses.

Abstract

The intrinsic timed-Hausdorff distance between timed-metric-spaces, first introduced by Sakovich--Sormani, yields a weak notion of convergence for space-times. In this paper we prove a compactness theorem for the intrinsic timed-Hausdorff convergence of timed-metric-spaces using timed-Fréchet maps. Our proof introduces the notion of ``addresses'' and provides a new way of stating Gromov's original compactness theorem for Gromov--Hausdorff convergence of metric spaces. We also obtain a new Arzelà--Ascoli theorem for real valued uniformly bounded Lipschitz functions on Gromov--Hausdorff converging compact metric spaces.
Paper Structure (17 sections, 11 theorems, 164 equations, 2 figures)

This paper contains 17 sections, 11 theorems, 164 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Review of the Hausdorff Distance
  4. Review of Gromov--Hausdorff Distance
  5. Review of Gromov's Compactness Theorem
  6. Review of Fréchet Maps
  7. Review of Timed-Metric-Spaces
  8. Review of Timed-Fréchet Maps
  9. Review of the Intrinsic Timed-Hausdorff Distance
  10. Proving the Compactness Theorem with Addresses
  11. Selection of Balls in a Compact Metric Space
  12. The Index Set of Addresses
  13. Compactness Theorem with Addresses
  14. Applications of Addresses
  15. Gromov's Compactness Theorem with Addresses
  16. ...and 2 more sections

Key Result

Theorem 1.1

If $(X_j,d_j,\tau_j)$ is a sequence of compact timed metric spaces that are equibounded, equicompact, and have uniform bounds on their Lipschitz one functions then a subsequence converges in the intrinsic timed-Hausdorff sense to a compact timed metric space $(X_\infty, d_\infty, \tau_\infty)$. In fact, there exist distance and time preserving timed-Fréchet maps, where $Z$ is compact, such tha

Figures (2)

  • Figure 3.1: An index set with $N(1)=1, N(2)=3, N(3)=4$ and the corresponding points in $X_j$.
  • Figure 3.2: An address $\alpha$ identifying a sequence of points in $X_j$ approaching the point ${\mathcal{I}}^j(\alpha)$.

Theorems & Definitions (23)

  • Theorem 1.1: Timed Gromov Compactness Theorem--short version
  • Theorem 2.1: Gromov Compactness Theorem
  • Definition 2.2
  • Theorem 2.3: Fréchet
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Definition 2.9
  • ...and 13 more