Duke Mathematical Journal
- Duke Math. J.
- Volume 151, Number 3 (2010), 431-485.
Regularity of optimal transport in curved geometry: The nonfocal case
We explore some geometric and analytic consequences of a curvature condition introduced by Ma, Trudinger, and Wang in relation to the smoothness of optimal transport in curved geometry. We discuss a conjecture according to which a strict version of the Ma-Trudinger-Wang condition is sufficient to prove regularity of optimal transport on a Riemannian manifold. We prove this conjecture under a somewhat restrictive additional assumption of nonfocality; at the same time, we establish the striking geometric property that the tangent cut locus is the boundary of a convex set. Partial extensions are presented to the case when there is no pure focalization on the tangent cut locus
Duke Math. J. Volume 151, Number 3 (2010), 431-485.
First available in Project Euclid: 8 February 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35J60: Nonlinear elliptic equations
Secondary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 49Q20: Variational problems in a geometric measure-theoretic setting
Loeper, Grégoire; Villani, Cédric. Regularity of optimal transport in curved geometry: The nonfocal case. Duke Math. J. 151 (2010), no. 3, 431--485. doi:10.1215/00127094-2010-003. https://projecteuclid.org/euclid.dmj/1265637659