Synthetic versus distributional lower Ricci curvature bounds
Synthetic versus distributional lower Ricci curvature bounds
We compare two standard approaches to defining lower Ricci curvature bounds for Riemannian metrics of regularity below C2. These are, on the one hand, the synthetic definition via weak displacement convexity of entropy functionals in the framework of optimal transport, and the distributional one based on non-negativity of the Ricci-tensor in the sense of Schwartz. It turns out that distributional bounds imply entropy bounds for metrics of class C1 and that the converse holds for C1,1-metrics under an additional convergence condition on regularizations of the metric.
Ricci curvature bounds, low regularity, optimal transport, synthetic geometry, tensor distributions
Kunzinger, Michael
5ee9f681-a923-4fb5-b1c8-9454237dd721
Oberguggenberger, Michael
45e9aedf-938e-4c76-ada3-b6d16b49e89b
Vickers, James A.
719cd73f-c462-417d-a341-0b042db88634
Kunzinger, Michael
5ee9f681-a923-4fb5-b1c8-9454237dd721
Oberguggenberger, Michael
45e9aedf-938e-4c76-ada3-b6d16b49e89b
Vickers, James A.
719cd73f-c462-417d-a341-0b042db88634
Kunzinger, Michael, Oberguggenberger, Michael and Vickers, James A.
(2023)
Synthetic versus distributional lower Ricci curvature bounds.
Proceedings of the Royal Society of Edinburgh Section A Mathematics.
(doi:10.1017/prm.2023.70).
Abstract
We compare two standard approaches to defining lower Ricci curvature bounds for Riemannian metrics of regularity below C2. These are, on the one hand, the synthetic definition via weak displacement convexity of entropy functionals in the framework of optimal transport, and the distributional one based on non-negativity of the Ricci-tensor in the sense of Schwartz. It turns out that distributional bounds imply entropy bounds for metrics of class C1 and that the converse holds for C1,1-metrics under an additional convergence condition on regularizations of the metric.
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Accepted/In Press date: 17 July 2023
e-pub ahead of print date: 23 August 2023
Additional Information:
Funding Information:
We thank Christian Ketterer for helpful discussions. This work was supported by project P 33594 of the Austrian Science Fund FWF.
Keywords:
Ricci curvature bounds, low regularity, optimal transport, synthetic geometry, tensor distributions
Identifiers
Local EPrints ID: 484977
URI: http://eprints.soton.ac.uk/id/eprint/484977
ISSN: 0308-2105
PURE UUID: 644671ee-fb49-4fa8-a454-30272047ec7b
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Date deposited: 27 Nov 2023 17:43
Last modified: 18 Mar 2024 02:32
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Author:
Michael Kunzinger
Author:
Michael Oberguggenberger
Author:
James A. Vickers
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