Abstract
These notes record the six lectures for the CIME Summer Course held by the second author in Cetraro during the week of June 23–28, 2008, with minor modifications. Their goal is to describe some recent developments in the theory of optimal transport, and their applications to differential geometry.
Keywords
- Riemannian Manifold
- Sectional Curvature
- Ricci Curvature
- Length Space
- Compact Riemannian Manifold
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References
L. Ambrosio, P. Tilli, Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and its Applications, vol. 25 (Oxford University Press, Oxford, 2004)
A.-I. Bonciocat, K.-T. Sturm, Mass transportation and rough curvature bounds for discrete spaces. J. Funct. Anal. 256(9), 2944–2966 (2009)
Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44(4), 375–417 (1991)
D. Burago, Y. Burago, S. Ivanov, A course in metric geometry. Graduate Studies in Mathematics, vol. 33 (American Mathematical Society, Providence, RI, 2001)
L.A. Caffarelli, The regularity of mappings with a convex potential. J. Am. Math. Soc. 5(1), 99–104 (1992)
L.A. Caffarelli, Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45(9), 1141–1151 (1992)
L.A. Caffarelli, Boundary regularity of maps with convex potentials. II. Ann. Math. 2 144(3), 453–496 (1996)
D. Cordero-Erausquin, Sur le transport de mesures périodiques. C. R. Acad. Sci. Paris Sèr. I Math. 329(3), 199–202 (1999)
D. Cordero-Erasquin, R.J. McCann, M. Schmuckenschlager, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146(2), 219–257 (2001)
P. Delanoë, Classical solvability in dimension two of the second boundary-value problem associated with the Monge–Ampère operator. Ann. Inst. H. Poincaré Anal. Non Linéaire 8(5), 443–457 (1991)
A. Fathi, A. Figalli, Optimal transportation on non-compact manifolds. Israel J. Math. 175(1), 1–59 (2010)
A. Figalli, L. Rifford, Continuity of optimal transport maps on small deformations of \({\mathbb{S}}^{2}\). Comm. Pure Appl. Math. 62(12), 1670–1706 (2009)
A. Figalli, C. Villani, An approximation lemma about the cut locus, with applications in optimal transport theory. Methods Appl. Anal. 15(2), 149–154 (2008)
W. Gangbo, R.J. McCann, The geometry of optimal transportation. Acta Math. 177(2), 113–161 (1996)
R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)
A. Joulin, A new Poisson-type deviation inequality for Markov jump process with positive Wasserstein curvature. Bernoulli 15(2), 532–549 (2009)
L.V. Kantorovich, On mass transportation. C. R. (Doklady) Acad. Sci. URSS (N.S.) (Reprinted) 37, 7-8 (1942)
L.V. Kantorovich, On a problem of Monge. C. R. (Doklady) Acad. Sci. URSS (N.S.) (Reprinted) 3, 2 (1948)
Y.H. Kim, Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds. Int. Math. Res. Not. IMRN, Art. ID rnn120, p.15 (2008)
Y.H. Kim, R.J. McCann, Continuity, curvature, and the general covariance of optimal transportation. J. Eur. Math. Soc. (JEMS) 12, 1009–1040 (2010)
G. Loeper, On the regularity of solutions of optimal transportation problems. Acta Math. 202(2), 241–283 (2009)
G. Loeper, Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna. Arch. Ration. Mech. Anal. (to appear)
G. Loeper, C. Villani, Regularity of optimal transport in curved geometry: the nonfocal case. Duke Matk. J. 151(3), 431–485 (2010)
J. Lott, Optimal transport and Perelman’s reduced volume. Calc. Var. Part. Differ. Equat. 36(1), 49–84 (2009)
J. Lott, C. Villani, Ricci curvature via optimal transport. Ann. Math. 169, 903–991 (2009)
J. Lott, C. Villani, Weak curvature conditions and functional inequalities. J. Funct. Anal. 245(1), 311–333 (2007)
X.N. Ma, N.S. Trudinger, X.J. Wang, Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177(2), 151–183 (2005)
R.J. McCann, P. Topping, Ricci flow, entropy and optimal transportation. Amer. J. Math. 132, 711–730 (2010)
R.J. McCann, Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(3), 589–608 (2001)
S.-I. Ohta, Finsler interpolation inequalities. Calc. Var. Part. Differ. Equat. 36, 211–249 (2009)
Y. Ollivier, Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256(3), 810–864 (2009)
F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Comm. Part. Differ. Equat. 26(1-2), 101–174 (2001)
F. Otto, C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2), 361–400 (2000)
M.-K. von Renesse, K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl. Math. 58(7), 923–940 (2005)
K.-T. Sturm, On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006)
K.-T. Sturm, On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006)
P. Topping, L-optimal transportation for Ricci flow. J. Reine Angew. Math. 636, 93–122 (2009)
J.I.E. Urbas, Regularity of generalized solutions of Monge–Ampére equations. Math. Z. 197(3), 365–393 (1988)
J. I. E. Urbas: On the second boundary value problem for equations of Monge–Ampère type. J. Reine Angew. Math. 487, 115–124 (1997)
C. Villani, Optimal transport, old and new. Notes for the 2005 Saint-Flour summer school. To appear in Grundlehren der mathematischen Wissenschaften. Preliminary version available at www.umpa.ens-lyon.fr/ ̃cvillani
C. Villani, Stability of a 4th-order curvature condition arising in optimal transport theory. J. Funct. Anal. 255(9), 2683–2708 (2008)
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Figalli, A., Villani, C. (2011). Optimal Transport and Curvature. In: Nonlinear PDE’s and Applications. Lecture Notes in Mathematics(), vol 2028. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21861-3_4
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