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arXiv · 2510.13069

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

Mauricio Che·Raquel Perales·Christina Sormani
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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-Fr\'echet 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 Arzel\`a--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.

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Mauricio Che, Raquel Perales, Christina Sormani. 2025-10-15. Gromov's Compactness Theorem for the Intrinsic Timed-Hausdorff Distance. https://arxiv.org/abs/2510.13069

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