Abstract
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: M → M pushing forward μ to a measure ν: each S factors uniquely a.e. into the composition S = T ◦ U, where U: M → M is volume preserving and T: M → M 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 Sn. 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.
Similar content being viewed by others
References
J. D. Benamou and Y. Brenier, A computational Fluid Mechanics solution to the Monge–Kantorovich mass transfer problem, Numerische Mathematik 84 (2000), 375–393.
L. Brasco, A survey on dynamical transport distances, Journal of Mathematical Sciences (New York) 181 (2012), 755–781.
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.
G. Buttazzo, Evolution models for mass transportation problems, Milan Journal of Mathematics 80 (2012), 47–63.
D. Cordero-Erausquin, Sur le transport de mesures périodiques, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 329 (1999), 199–202.
D. Cordero-Erausquin, R. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Inventiones Mathematicae 146 (2001), 219–257.
D. Emmanuele and M. Salvai, Force free Möbius motions of the circle, Journal of Geometry and Symmetry in Physics 27 (2012), 59–65.
U. Hertrich-Jeromin, Introduction to Möbius Differential Geometry, London Mathematical Society Lecture Note Series, Vol. 300, Cambridge University Press, Cambridge, 2003.
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.
Y. H. Kim, R. McCann and M. Warren, Pseudo-Riemannian geometry calibrates optimal transportation, Mathematical Research Letters 17 (2010), 1183–1197.
M. M. Lazarte, M. Salvai and A. Will, Force free projective motions of the sphere, Journal of Geometry and Physics 57 (2007), 2431–2436.
N. Q. Le, Hölder regularity of the 2D dual semigeostrophic equations via analysis of linearized Monge–Ampère equations, Communications in Mathematical Physics, to appear.
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.
R. McCann, Polar factorization of maps on Riemannian manifolds, Geometric and Functional Analysis 11 (2001), 589–608.
M. Salvai, Force free conformal motions of the sphere, Differential Geometry and its Applications 16 (2002), 285–292.
C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, Vol. 338, Springer-Verlag, Berlin, 2009.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by Conicet (PIP 112-2011-01-00670), Foncyt (PICT 2010 cat 1 proyecto 1716) Secyt Univ. Nac. Córdoba.
Rights and permissions
About this article
Cite this article
Godoy, Y., Salvai, M. Polar factorization of conformal and projective maps of the sphere in the sense of optimal mass transport. Isr. J. Math. 225, 465–478 (2018). https://doi.org/10.1007/s11856-018-1673-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-018-1673-5