Lorentzian Gromov–Hausdorff convergence and pre-compactness
- 1Mathematical Institute , University of Oxford , Radcliffe Observatory, Andrew Wiles Building, Woodstock Rd , Oxford , OX2 6GG , United Kingdom
- 2Faculty of Mathematics , University of Vienna , Oskar-Morgenstern-Platz 1, 1090 Vienna , Austria
Abstract
The goal of this paper is to introduce a notion of convergence à la Gromov–Hausdorff for Lorentzian spaces, based on 𝜀-nets formed by causal diamonds and depending solely on the time-separation function. This provides a geometric framework for convergence that applies both to synthetic Lorentzian spaces (Lorentzian pre-length spaces) and to smooth spacetimes. Among our main results, we establish a Lorentzian analogue of Gromov’s celebrated pre-compactness theorem for metric spaces, where controlled covers by balls are replaced with controlled covers by diamonds. This leads to two geometric pre-compactness results for families of globally hyperbolic 𝑛-dimensional spacetimes ( M n , g ) (M^{n},g) . The first relies on suitable causality control together with the existence of a Cauchy hypersurface Σ satisfying a uniform doubling property. The second assumes the existence of a compact Cauchy surface Σ with bounded second fundamental form and bounded diameter; a lower bound on the ambient Ricci curvature along Σ; a lower bound on timelike sectional curvature on 𝑀. This is the first Lorentzian pre-compactness result obtained under curvature–diameter assumptions. In the final part of the paper, we present several applications. We show that Chruściel–Grant approximations [P. T. Chruściel and J. D. E. Grant, On Lorentzian causality with continuous metrics, Classical Quantum Gravity 29 (2012), 14, Article ID 145001] are special cases of the Lorentzian Gromov–Hausdorff convergence introduced here, prove that timelike sectional curvature bounds are stable under this convergence, introduce timelike blow-up tangents, and discuss connections with the main conjecture of causal set theory.
Related
- Gromov-Hausdorff Convergence of Spectral Truncations for Noncommutative Tori
- Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class
- Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds
- Gromov-Hausdorff instability of the positive mass theorem for Kähler manifolds
- Pointed Gromov-Hausdorff Distance between lp-Wedge Sums and Isometric Embeddings of l∞
BibTeX
@article{mondino2026lorentzian,
title = {{Lorentzian Gromov–Hausdorff convergence and pre-compactness}},
author = {Andrea Mondino and Clemens Sämann},
journal = {Journal für die reine und angewandte Mathematik (Crelles Journal)},
year = {2026},
month = aug,
doi = {10.1515/crelle-2026-0067},
url = {https://doi.org/10.1515/crelle-2026-0067},
note = {CC BY},
}
RIS
TY - JOUR
TI - Lorentzian Gromov–Hausdorff convergence and pre-compactness
AU - Andrea Mondino
AU - Clemens Sämann
PY - 2026
DA - 2026/08/28
JO - Journal für die reine und angewandte Mathematik (Crelles Journal)
DO - 10.1515/crelle-2026-0067
UR - https://doi.org/10.1515/crelle-2026-0067
AB - The goal of this paper is to introduce a notion of convergence à la Gromov–Hausdorff for Lorentzian spaces, based on 𝜀-nets formed by causal diamonds and depending solely on the time-separation function. This provides a geometric framework for convergence that applies both to synthetic Lorentzian spaces (Lorentzian pre-length spaces) and to smooth spacetimes. Among our main results, we establish a Lorentzian analogue of Gromov’s celebrated pre-compactness theorem for metric spaces, where controlled covers by balls are replaced with controlled covers by diamonds. This leads to two geometric pre-compactness results for families of globally hyperbolic 𝑛-dimensional spacetimes ( M n , g ) (M^{n},g) . The first relies on suitable causality control together with the existence of a Cauchy hypersurface Σ satisfying a uniform doubling property. The second assumes the existence of a compact Cauchy surface Σ with bounded second fundamental form and bounded diameter; a lower bound on the ambient Ricci curvature along Σ; a lower bound on timelike sectional curvature on 𝑀. This is the first Lorentzian pre-compactness result obtained under curvature–diameter assumptions. In the final part of the paper, we present several applications. We show that Chruściel–Grant approximations [P. T. Chruściel and J. D. E. Grant, On Lorentzian causality with continuous metrics, Classical Quantum Gravity 29 (2012), 14, Article ID 145001] are special cases of the Lorentzian Gromov–Hausdorff convergence introduced here, prove that timelike sectional curvature bounds are stable under this convergence, introduce timelike blow-up tangents, and discuss connections with the main conjecture of causal set theory.
SN - 0075-4102
SN - 1435-5345
ER -