Duke Mathematical Journal

Regularity of optimal transport in curved geometry: The nonfocal case

Grégoire Loeper and Cédric Villani

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Abstract

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

Source
Duke Math. J. Volume 151, Number 3 (2010), 431-485.

Dates
First available in Project Euclid: 8 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1265637659

Digital Object Identifier
doi:10.1215/00127094-2010-003

Mathematical Reviews number (MathSciNet)
MR2605867

Zentralblatt MATH identifier
1192.53041

Subjects
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

Citation

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


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References

  • 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.