Israel Journal of Mathematics

, Volume 225, Issue 1, pp 465–478 | Cite as

Polar factorization of conformal and projective maps of the sphere in the sense of optimal mass transport

  • Yamile Godoy
  • Marcos Salvai


Let M be a compact Riemannian manifold and let μ, d be the associated measure and distance on M. Robert McCann, generalizing results for the Euclidean case by Yann Brenier, obtained the polar factorization of Borel maps S: MM pushing forward μ to a measure ν: each S factors uniquely a.e. into the composition S = TU, where U: MM is volume preserving and T: MM is the optimal map transporting μ to ν with respect to the cost function d2/2.

In this article we study the polar factorization of conformal and projective maps of the sphere S n . For conformal maps, which may be identified with elements of Oo(1, n+1), we prove that the polar factorization in the sense of optimal mass transport coincides with the algebraic polar factorization (Cartan decomposition) of this Lie group. For the projective case, where the group GL+(n + 1) is involved, we find necessary and sufficient conditions for these two factorizations to agree.


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  1. [1]
    J. D. Benamou and Y. Brenier, A computational Fluid Mechanics solution to the Monge–Kantorovich mass transfer problem, Numerische Mathematik 84 (2000), 375–393.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    L. Brasco, A survey on dynamical transport distances, Journal of Mathematical Sciences (New York) 181 (2012), 755–781.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, Comptes Rendus des Séances de l’Académie des Sciences. Série I. Mathématique 305 (1987), 805–808.zbMATHGoogle Scholar
  4. [4]
    G. Buttazzo, Evolution models for mass transportation problems, Milan Journal of Mathematics 80 (2012), 47–63.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D. Cordero-Erausquin, Sur le transport de mesures périodiques, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 329 (1999), 199–202.MathSciNetzbMATHGoogle Scholar
  6. [6]
    D. Cordero-Erausquin, R. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Inventiones Mathematicae 146 (2001), 219–257.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    D. Emmanuele and M. Salvai, Force free Möbius motions of the circle, Journal of Geometry and Symmetry in Physics 27 (2012), 59–65.MathSciNetzbMATHGoogle Scholar
  8. [8]
    U. Hertrich-Jeromin, Introduction to Möbius Differential Geometry, London Mathematical Society Lecture Note Series, Vol. 300, Cambridge University Press, Cambridge, 2003.Google Scholar
  9. [9]
    Y. H. Kim and R. McCann, Continuity, curvature, and the general covariance of optimal transportation, Journal of the European Mathematical Society 12 (2010), 1009–1040.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Y. H. Kim, R. McCann and M. Warren, Pseudo-Riemannian geometry calibrates optimal transportation, Mathematical Research Letters 17 (2010), 1183–1197.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. M. Lazarte, M. Salvai and A. Will, Force free projective motions of the sphere, Journal of Geometry and Physics 57 (2007), 2431–2436.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    N. Q. Le, Hölder regularity of the 2D dual semigeostrophic equations via analysis of linearized Monge–Ampère equations, Communications in Mathematical Physics, to appear.Google Scholar
  13. [13]
    G. Loeper, Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna, Archive for Rational Mechanics and Analysis 199 (2011), 269–289.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    R. McCann, Polar factorization of maps on Riemannian manifolds, Geometric and Functional Analysis 11 (2001), 589–608.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    M. Salvai, Force free conformal motions of the sphere, Differential Geometry and its Applications 16 (2002), 285–292.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, Vol. 338, Springer-Verlag, Berlin, 2009.Google Scholar

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© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.CIEM - FaMAFConicet - Universidad Nacional de Córdoba, Ciudad UniversitariaCórdobaArgentina

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