Synthetic Curvature from Optimal Transport on Metric Measure Spaces Article Swipe
Revista, Zen
,
MATH, 10
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.5281/zenodo.17838997
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.5281/zenodo.17838997
This paper explores the construction and implications of synthetic notions of Ricci curvature on general metric measure spaces through the lens of optimal transport theory. Traditional differential geometric definitions of curvature rely on a smooth manifold structure, which is absent in many contemporary mathematical and physical settings. Optimal transport provides a robust framework to extend these fundamental geometric concepts to non-smooth spaces, characterized only by a distance function and a measure. We delve into the foundational Lott-Sturm-Villani (LSV) theory, which defines curvature-dimension conditions, such as CD(K,N) and RCD(K,N), through the convexity properties of entropy functionals along Wasserstein geodesics. The equivalence of these synthetic conditions with classical Bakry-Émery curvature bounds in smooth settings is discussed, highlighting the unifying power of optimal transport. We examine the stability of these conditions under various forms of convergence and their profound implications for functional inequalities like the Bishop-Gromov, Brunn-Minkowski, and HWI inequalities. Furthermore, the paper touches upon applications ranging from eigenvalue bounds in non-smooth settings to connections with gravitational theories and diffusion processes, demonstrating the broad utility and fundamental nature of synthetic curvature derived from optimal transport.
Related Topics
Concepts
Mathematics
Curvature
Ricci curvature
Measure (data warehouse)
Manifold (fluid mechanics)
Probability measure
Convexity
Metric (unit)
Differential geometry
Eigenvalues and eigenvectors
Stability (learning theory)
Equivalence (formal languages)
Entropy (arrow of time)
Metric space
Convergence (economics)
Mean curvature flow
Scalar curvature
Mathematical analysis
Function (biology)
Function space
Applied mathematics
Differential (mechanical device)
Wasserstein metric
Pure mathematics
Mean curvature
Term (time)
Riemannian manifold
Statistical manifold
Convex function
Metadata
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- article
- Landing Page
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- OpenAlex ID
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All OpenAlex metadata
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https://openalex.org/W7109726976Canonical identifier for this work in OpenAlex
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https://doi.org/10.5281/zenodo.17838997Digital Object Identifier
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Synthetic Curvature from Optimal Transport on Metric Measure SpacesWork title
- Type
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articleOpenAlex work type
- Publication year
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2025Year of publication
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2025-12-06Full publication date if available
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Revista, Zen, MATH, 10List of authors in order
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https://doi.org/10.5281/zenodo.17838997Publisher landing page
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
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https://doi.org/10.5281/zenodo.17838997Direct OA link when available
- Concepts
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Mathematics, Curvature, Ricci curvature, Measure (data warehouse), Manifold (fluid mechanics), Probability measure, Convexity, Metric (unit), Differential geometry, Eigenvalues and eigenvectors, Stability (learning theory), Equivalence (formal languages), Entropy (arrow of time), Metric space, Convergence (economics), Mean curvature flow, Scalar curvature, Mathematical analysis, Function (biology), Function space, Applied mathematics, Differential (mechanical device), Wasserstein metric, Pure mathematics, Mean curvature, Term (time), Riemannian manifold, Statistical manifold, Convex functionTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile |