Metric currents and geometry of Wasserstein spaces
Rendiconti del Seminario Matematico della Università di Padova, Volume 124 (2010), pp. 91-125.
@article{RSMUP_2010__124__91_0,
     author = {Granieri, Luca},
     title = {Metric currents and geometry of {Wasserstein} spaces},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {91--125},
     publisher = {Seminario Matematico of the University of Padua},
     volume = {124},
     year = {2010},
     mrnumber = {2752678},
     zbl = {1210.35076},
     language = {en},
     url = {http://archive.numdam.org/item/RSMUP_2010__124__91_0/}
}
TY  - JOUR
AU  - Granieri, Luca
TI  - Metric currents and geometry of Wasserstein spaces
JO  - Rendiconti del Seminario Matematico della Università di Padova
PY  - 2010
SP  - 91
EP  - 125
VL  - 124
PB  - Seminario Matematico of the University of Padua
UR  - http://archive.numdam.org/item/RSMUP_2010__124__91_0/
LA  - en
ID  - RSMUP_2010__124__91_0
ER  - 
%0 Journal Article
%A Granieri, Luca
%T Metric currents and geometry of Wasserstein spaces
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2010
%P 91-125
%V 124
%I Seminario Matematico of the University of Padua
%U http://archive.numdam.org/item/RSMUP_2010__124__91_0/
%G en
%F RSMUP_2010__124__91_0
Granieri, Luca. Metric currents and geometry of Wasserstein spaces. Rendiconti del Seminario Matematico della Università di Padova, Volume 124 (2010), pp. 91-125. http://archive.numdam.org/item/RSMUP_2010__124__91_0/

[1] L. Ambrosio, Lecture Notes on Transport Problems, in "Mathematical Aspects of Evolving Interfaces". Lecture Notes in Mathematics, 1812 (Springer, Berlin, 2003), pp. 1--52. | MR | Zbl

[2] L. Ambrosio - N. Gigli - G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 2005. | MR | Zbl

[3] L. Ambrosio - B. Kirchheim, Currents in Metric Spaces, Acta Mathematica, 185, no. 1 (2000), pp. 1--80. | MR | Zbl

[4] L. Ambrosio - P. Tilli, Topics on Analysis in Metric Spaces, Oxford Lectures Series in Mathematics and its Applications, 25, Oxford University Press, Oxford, 2004. | MR | Zbl

[5] V. Bangert, Minimal measures and minimizing closed normal one-currents, GAFA Geom. Funct. Anal., 9, no. 3 (1999), pp. 413--427. | MR | Zbl

[6] G. Bouchitté - G. Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation, Journal European Math. Soc., 3 (2001), pp. 139--168. | MR | Zbl

[7] Y. Brenier, Extended Monge-Kantorovich Theory, in Optimal Transportation and Applications, Lecture Notes in Mathematics, 1813 (Springer, Berlin, 2003), pp. 91--121. | MR | Zbl

[8] J. Benamou - Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84, no. 3 (2000), pp. 375--393. | MR | Zbl

[9] P. Bernard - B. Buffoni, Optimal mass transportation and Mather theory, J. Eur. Math. Soc. (JEMS), 9, no. 1 (2007), pp. 85--121. | MR

[10] G. Contreras - R. Iturriaga, Global Minimizers of Autonomous Lagrangians. IMPA, Rio de Janeiro, 1999. | MR | Zbl

[11] L. De Pascale - M. S. Gelli - L. Granieri, Minimal measures, one-dimensional currents and the Monge-Kantorovich problem, Calc. Var., 27, no. 1 (2006), pp. 1--23. | MR | Zbl

[12] B. Dacorogna, Direct Methods in the Calculus of Variations, second edition, Springer, 2008. | MR | Zbl

[13] L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer (survey paper), Current Developments in Mathematics, 1997, International Press (1999), edited by S. T. Yau. | MR | Zbl

[14] W. Fulton, Algebraic Topology. Springer, 1995. | MR | Zbl

[15] H. Federer, Geometric Measure Theory. Springer (Berlin), 1969. | MR | Zbl

[16] W. Gangbo - H. Kil - T. Pacini, Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems, forthcoming on Memoirs AMS. | Zbl

[17] W. Gangbo - R. J. Mc Cann, The geometry of optimal transportation, Acta Math., 177 (1996), pp. 113--161. | MR | Zbl

[18] M. Giaquinta - G. Modica - J. Souček, Cartesian currents in the calculus of variations. I. Cartesian currents. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 37. Springer-Verlag, Berlin, 1998. | MR | Zbl

[19] L. Granieri, On action minimizing measures for the Monge-Kantorovich problem, NoDEA 14 (2007), pp. 125--152. | MR | Zbl

[20] L. Granieri, Mass Transportation Problems and Minimal Measures. Ph.D. Thesis in Mathematics, Pisa, 2005.

[21] B. Kloeckner, Geometric study of Wasserstein spaces: Euclidean spaces, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze IX, 2 (2010), pp. 297--323. | Numdam | MR | Zbl

[22] R. Jordan - D. Kinderlehrer - F. Otto, The variational formulation of the Fokker-Plank equation, Siam J. Math. Anal., 29 (1998), pp. 1--17. | MR | Zbl

[23] J. Jost, Riemannian Geometry and Geometric Analysis. Springer, 2002. | MR | Zbl

[24] J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects. Lectures in Math. ETH Zurich, Birkhauser Verlag, Basel, 1997. | MR | Zbl

[25] U. Lang - V. Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature, GAFA Geom. Funct. Anal., 7 (1997), pp. 535--560. | MR | Zbl

[26] F. Otto, The geometry of dissipative evolution equations: the porus medium equation, Comm. Partial Differential Equations 26, no. 1-2 (2001), pp. 101--174. | MR | Zbl

[27] K. T. Sturm, Stochastics and Analysis on Metric Spaces, lecture notes in preparation.

[28] K. T. Sturm, Metric spaces of lower bounded curvature, Exposition. Math., 17, no. 1 (1999), pp. 35--47. | MR | Zbl

[29] K. T. Sturm, Probability Measures on Metric Spaces of Nonpositive Curvature, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), pp. 357--390, Contemp. Math., 338, AMS, Providence, RI, 2003. | MR | Zbl

[30] K. T. Sturm, On the geometry of metric measure spaces. I, Acta Math., 196, no.1 (2006), pp. 65--131. | MR | Zbl

[31] C. Villani, Topics in Mass Transportation. Graduate Studies in mathematics, 58, AMS, Providence, RI, 2003. | MR | Zbl

[32] C. Villani, Optimal Transport, Old and New. Springer, 2009. | MR | Zbl

[33] S. Wenger, Isoperimetric inequalities of euclidean type in metric spaces, GAFA, Geom. funct. anal., Vol. 15 (2005), pp. 534--554. | MR | Zbl

[34] S. Wenger, Flat convergence for integral currents in metric spaces, Calc. Var., 28 (2007), pp. 139--160. | MR | Zbl