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Abstract We first provide a stochastic formula for the Carathéodory distance in terms of general Markovian couplings and prove a comparison result between the Carathéodory distance and the complete Kähler metric with a negative lower curvature bound using the Kendall–Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete noncompact Kähler manifolds with a further decomposition Ricci curvature into the orthogonal Ricci curvature and the holomorphic sectional curvature, which cannot be obtained by using Yau–Royden's Schwarz lemma. We also prove coupling estimates on quaternionic Kähler manifolds. As a by‐product, we obtain an improved gradient estimate of positive harmonic functions on Kähler manifolds and quaternionic Kähler manifolds under lower curvature bounds.
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A Toponogov globalisation result for Lorentzian length spaces
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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- Award ID(s):
- 1928930
- PAR ID:
- 10588606
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Mathematische Annalen
- Volume:
- 392
- Issue:
- 3
- ISSN:
- 0025-5831
- Format(s):
- Medium: X Size: p. 3447-3478
- Size(s):
- p. 3447-3478
- Sponsoring Org:
- National Science Foundation
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