A Toponogov globalisation result for Lorentzian length spaces.

IF 1.4 2区 数学 Q1 MATHEMATICS
Mathematische Annalen Pub Date : 2025-01-01 Epub Date: 2025-05-07 DOI:10.1007/s00208-025-03167-w
Tobias Beran, John Harvey, Lewis Napper, Felix Rott
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引用次数: 0

Abstract

In the synthetic geometric setting introduced by Kunzinger and Sämann, we present an analogue of Toponogov's Globalisation Theorem which applies to Lorentzian length spaces with lower (timelike) curvature bounds. Our approach utilises a "cat's cradle" construction akin to that which appears in several proofs in the metric setting. On the road to our main result, we also provide a lemma regarding the subdivision of triangles in spaces with a local lower curvature bound and a synthetic Lorentzian version of the Lebesgue Number Lemma. Several properties of time functions and the null distance on globally hyperbolic Lorentzian length spaces are also highlighted. We conclude by presenting several applications of our results, including versions of the Bonnet-Myers Theorem and the Splitting Theorem for Lorentzian length spaces with local lower curvature bounds, as well as discussion of stability of curvature bounds under Gromov-Hausdorff convergence.

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洛伦兹长度空间的Toponogov全球化结果。
在Kunzinger和Sämann引入的合成几何设置中,我们给出了Toponogov全球化定理的一个模拟,该定理适用于具有较低(类时)曲率界的洛伦兹长度空间。我们的方法利用了“猫的摇篮”结构,类似于在公制设置的几个证明中出现的结构。在通往我们的主要结果的道路上,我们还提供了一个关于局部下曲率界空间中三角形细分的引理和勒贝格数引理的合成洛伦兹版本。讨论了时间函数和零距离在全局双曲洛伦兹长度空间上的几个性质。最后,我们给出了我们的结果的几个应用,包括具有局部下曲率界的洛伦兹长度空间的Bonnet-Myers定理和分裂定理的版本,以及讨论了Gromov-Hausdorff收敛下曲率界的稳定性。
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来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
期刊介绍: Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.
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