Mass Transportation on Sub-Riemannian Manifolds

2008/03/20 by Alessio Figalli, Figalli, Alessio, Ludovic Rifford +1 · 3 citations
Computer Science · Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Optimization and Control (math.OC) #Topological and Geometric Data Analysis

paper · doi:10.48550/arxiv.0803.2917

openalex publication_date 2008/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a weak form of the Monge-Ampère equation.

Citations

• Shortest paths for sub-Riemannian metrics on rank-two distributions
• Polar factorization and monotone rearrangement of vector‐valued functions

Cited by

• Monge solutions and uniqueness in multi-marginal optimal transport with hierarchical jumps
• Flow Matching for Measure Transport and Feedback Stabilization of Control-Affine Systems
• Benamou-Brenier and Kantorovich on sub-Riemannian manifolds with no abnormal geodesics

Related

• Mass Transportation on sub-Riemannian structures of rank two in dimension four
• Optimal transportation on non-compact manifolds
• Limit Theorems for Optimal Mass Transportation
• Multi-marginal optimal transport on Riemannian manifolds
• Optimal Transportation under Nonholonomic Constraints