Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)

DOI10.1515/CRELLE.2011.105zbMath1239.53049arXiv0806.0351OpenAlexW2963529094WikidataQ115236895 ScholiaQ115236895MaRDI QIDQ2879865

Robert J. McCann, Young-Heon Kim

Publication date: 5 April 2012

Published in: Journal für die reine und angewandte Mathematik (Crelles Journal) (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0806.0351

zbMATH Keywords

curvatureoptimal transportMa-Trudinger-Wang conditionRiemannian distance squared cost

Mathematics Subject Classification ID

Global Riemannian geometry, including pinching (53C20) Local differential geometry of Lorentz metrics, indefinite metrics (53B30) Existence theories in calculus of variations and optimal control (49J99)

Related Items (21)

Regularity of optimal transport maps on multiple products of spheres ⋮ On the Degeneracy of Optimal Transportation ⋮ The Monge–Ampère equation and its link to optimal transportation ⋮ A glimpse into the differential topology and geometry of optimal transport ⋮ On the Ma-Trudinger-Wang curvature on surfaces ⋮ Hölder continuity and injectivity of optimal maps ⋮ Positivity of Ma-Trudinger-Wang curvature on Riemannian surfaces ⋮ A perturbation argument for a Monge-Ampère type equation arising in optimal transportation ⋮ Bipolar comparison ⋮ Wasserstein barycenters over Riemannian manifolds ⋮ Regularity of optimal transport in curved geometry: the nonfocal case ⋮ The Ma-Trudinger-Wang curvature for natural mechanical actions ⋮ When is multidimensional screening a convex program? ⋮ Continuity and injectivity of optimal maps ⋮ Regularity of optimal transport maps on locally nearly spherical manifolds ⋮ A variational formulation of the BDF2 method for metric gradient flows ⋮ Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds ⋮ A Jacobian Inequality for Gradient Maps on the Sphere and Its Application to Directional Statistics ⋮ On the Convexity of Injectivity Domains on Nonfocal Manifolds ⋮ Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations ⋮ Partial regularity for optimal transport maps

Cites Work

• On the Ma-Trudinger-Wang curvature on surfaces
• On the regularity of solutions of optimal transportation problems
• Boundary regularity of maps with convex potentials. II
• Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna
• \(C^{1}\) regularity of solutions of the Monge-Ampère equation for optimal transport in dimension two
• On strict convexity and continuous differentiability of potential functions in optimal transportation
• Hedonic price equilibria, stable matching, and optimal transport: Equivalence, topology, and uniqueness
• Regularity of optimal transport in curved geometry: the nonfocal case
• Continuity, curvature, and the general covariance of optimal transportation
• Stability of a 4th-order curvature condition arising in optimal transport theory
• Hölder regularity of optimal mappings in optimal transportation
• Gradient rearrangement for diffeomorphisms of a compact manifold.
• Convex functionals of probability measures and nonlinear diffusions on manifolds
• Regularity of potential functions of the optimal transportation problem
• Gradient estimates for potentials of invertible gradient-mappings on the sphere
• InteriorC^2,αRegularity for Potential Functions in Optimal Transportation
• Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊²
• Polar factorization and monotone rearrangement of vector‐valued functions
• Existence, Uniqueness, and Regularity of Optimal Transport Maps
• Polar factorization of maps on Riemannian manifolds

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