Gromov's Compactness Theorem for the Intrinsic Timed-Hausdorff Distance
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917296394469376 |
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| author | Che, Mauricio Perales, Raquel Sormani, Christina |
| author_facet | Che, Mauricio Perales, Raquel Sormani, Christina |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gromov's Compactness Theorem for the Intrinsic Timed-Hausdorff Distance Che, Mauricio Perales, Raquel Sormani, Christina Metric Geometry Differential Geometry 53C24 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. |
| title | Gromov's Compactness Theorem for the Intrinsic Timed-Hausdorff Distance |
| topic | Metric Geometry Differential Geometry 53C24 |
| url | https://arxiv.org/abs/2510.13069 |