Duke Mathematical Journal

Regularity of optimal transport in curved geometry: The nonfocal case

Grégoire Loeper and Cédric Villani

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Article information

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.

Export citation


  • R. L. Bishop, Decomposition of cut loci, Proc. Amer. Math. Soc. 65 (1977), 133--136.
  • L. A. Caffarelli, Boundary regularity of maps with convex potentials, Comm. Pure Appl. Math. 45 (1992), 1141--1151.
  • —, The regularity of mappings with a convex potential, J. Amer. Math. Soc. 5 (1992), 99--104.
  • —, Boundary regularity of maps with convex potentials, II, Ann. of Math. (2) 144 (1996), 453--496.
  • D. Cordero-Erausquin, R. J. Mccann, and M. SchmuckenschläGer, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), 219--257.
  • Ph. Delanoë and Y. Ge, Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds, to appear in J. Reine Angew. Math.
  • Ph. Delanoë and G. Loeper, Gradient estimates for potentials of invertible gradient-mappings on the sphere, Calc. Var. Partial Differential Equations 26 (2006), 297--311.
  • M. P. Do Carmo, ``Riemannian geometry'' in Mathematics: Theory & Applications, Birkhaüser, Boston, 1992.
  • H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York, 1969.
  • A. Figalli and L. Rifford, Continuity of optimal transport maps on small deformations of $\Sph^2$, preprint, 2008.
  • A. Figalli and C. Villani, An approximation lemma about the cut locus, with applications in optimal transport theory, Methods Appl. Anal. 15 (2008), 149--154.
  • S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, 2nd ed., Universitext, Springer, Berlin, 1990.
  • J. I. Itoh and M. Tanaka, The Lipschitz continuity of the distance function to the cut locus, Trans. Amer. Math. Soc. 353 (2001), 21--40.
  • Y.-H. Kim, Counterexamples to continuity of optimal transport maps on positively curved Riemannian manifolds, Int. Math. Res. Not. IMRN (2008), art.IDrnn120.
  • Y.-H. Kim and R. J. Mccann, Continuity, curvature, and the general covariance of optimal transportation, to appear in J. Eur. Math. Soc., preprint.
  • —, On the cost-subdifferentials of cost-convex functions, preprint.
  • —, Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular), preprint.
  • Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math. 58 (2004), 85--146.
  • J. Liu, Hölder regularity of optimal mappings in optimal transportation, Calc. Var. Partial Differential Equations 34 (2009), 435--451.
  • G. Loeper, On the regularity of solutions of optimal transportation problems, Acta Math. 202 (2009), 241--283.
  • —, Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna, to appear in Arch. Ration. Mech. Anal.
  • X.-N. Ma, N. S. Trudinger, and X.-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal. 177 (2005), 151--183.
  • C. Mantegazza and A. C. Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Appl. Math. Optim. 47 (2003), 1--25.
  • R. J. Mccann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal. 11 (2001), 589--608.
  • L. Rifford, Notes on the local inversion theorem, personal communication, October 2007.
  • N. S. Trudinger and X.-J. Wang, On strict convexity and continuous differentiability of potential functions in optimal transportation, Arch. Ration. Mech. Anal. 192 (2009), 403--418.
  • —, On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), 143--174.
  • J. Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math. 487 (1997), 115--124.
  • C. Villani, Topics in optimal transportation Grad. Stud. in Math. 58, Amer. Math. Soc., Providence, 2003.
  • —, Stability of a 4th-order curvature condition arising in optimal transport theory, J. Funct. Anal. 255 (2008), 2683--2708.
  • —, Optimal Transport: Old and New, Grundlehren Math. Wiss. 338, Springer, Berlin, 2009.
  • G. T. Von Nessi, Regularity results for potential functions of the optimal transportation problem on spheres and related Hessian equations, Ph.D. dissertation, Australian National University, Canberra, Australia, 2008.
  • A. D. Weinstein, The cut locus and conjugate locus of a Riemannian manifold, Ann. of Math. (2) 87 (1968), 29--41.