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

2025/10/15 by Che, Mauricio, Che, M, Raquel Perales +3 · 2 citations
Mathematics · #53C24 #Advanced Topology and Set Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2510.13069

openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

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. Moreover, we establish the triangle inequality for the intrinsic timed-Hausdorff distance.

Citations

• Introducing Various Notions of Distances between Space-Times
• Gromov–Hausdorff convergence of metric pairs and metric tuples
• Null Distance and Gromov-Hausdorff Convergence of Warped Product Spacetimes
• Metric Geometry of Spaces of Persistence Diagrams
• Intrinsic Flat Arzela-Ascoli Theorems
• Lower Bounds on Ricci Curvature and Quantitative Behavior of Singular Sets
• The covering spectrum of a compact length space
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator

Cited by

• Intrinsic Timed Hausdorff Convergence and Its Implications
• Results on Lorentzian metric spaces
• Existence of uniform Temple charts and applications to null distance

Related

• Intrinsic Timed Hausdorff Convergence and Its Implications
• Convergence in the space of compact labeled metric spaces
• Gromov–Hausdorff convergence of metric pairs and metric tuples
• Lipschitz (non-)equivalence of the Gromov--Hausdorff distances, including on ultrametric spaces
• Realizations of Gromov-Hausdorff Distance