Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below - the Compact Case

Luigi Ambrosio; Nicola Gigli; Giuseppe Savarè

Bollettino dell'Unione Matematica Italiana (2012)

  • Volume: 5, Issue: 3, page 575-629
  • ISSN: 0392-4041

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Ambrosio, Luigi, Gigli, Nicola, and Savarè, Giuseppe. "Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below - the Compact Case." Bollettino dell'Unione Matematica Italiana 5.3 (2012): 575-629. <http://eudml.org/doc/290964>.

@article{Ambrosio2012,
author = {Ambrosio, Luigi, Gigli, Nicola, Savarè, Giuseppe},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {575-629},
publisher = {Unione Matematica Italiana},
title = {Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below - the Compact Case},
url = {http://eudml.org/doc/290964},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Ambrosio, Luigi
AU - Gigli, Nicola
AU - Savarè, Giuseppe
TI - Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below - the Compact Case
JO - Bollettino dell'Unione Matematica Italiana
DA - 2012/10//
PB - Unione Matematica Italiana
VL - 5
IS - 3
SP - 575
EP - 629
LA - eng
UR - http://eudml.org/doc/290964
ER -

References

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  1. AMBROSIO, L. - GIGLI, N., User's guide to optimal transport theory, to appear. MR3050280DOI10.1007/978-3-642-32160-3_1
  2. AMBROSIO, L. - GIGLI, N. - MONDINO, A. - SAVARÉ, G. - RAJALA, T., work in progress (2012). 
  3. AMBROSIO, L. - GIGLI, N. - SAVARÉ, G., Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008. Zbl1145.35001MR2401600
  4. AMBROSIO, L. - GIGLI, N. - SAVARÉ, G., Calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below, Arxiv 1106.2090, (2011), 1-74. MR3060497
  5. AMBROSIO, L. - GIGLI, N. - SAVARÉ, G. , Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Arxiv 1111.3730, (2011), 1-28. MR3090143DOI10.4171/RMI/746
  6. AMBROSIO, L. - GIGLI, N. - SAVARÉ, G. , Metric measure spaces with Riemannian Ricci curvature bounded from below, Arxiv 1109.0222, (2011), 1-60. MR3205729DOI10.1215/00127094-2681605
  7. AMBROSIO, L. - RAJALA, T., Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces, To appear on Ann. Mat. Pura Appl. (2012). MR3158838DOI10.1007/s10231-012-0266-x
  8. BRÉZIS, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). Zbl0252.47055
  9. CHEEGER, J., Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517. Zbl0942.58018MR1708448DOI10.1007/s000390050094
  10. DANERI, S. - SAVARÉ, G., Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40 (2008), 1104-1122. Zbl1166.58011MR2452882DOI10.1137/08071346X
  11. FUKUSHIMA, M., Dirichlet forms and Markov processes, vol. 23 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, 1980. Zbl0422.31007MR569058
  12. GIGLI, N., On the heat flow on metric measure spaces: existence, uniqueness and stability, Calc. Var. Partial Differential Equations, 39 (2010), 101-120. Zbl1200.35178MR2659681DOI10.1007/s00526-009-0303-9
  13. GIGLI, N., On the differential structure of metric measure spaces and applications, Submitted paper, (2012). Zbl1325.53054MR3381131DOI10.1090/memo/1113
  14. GIGLI, N., Optimal maps in non branching spaces with Ricci curvature bounded from below, To appear on Geom. Funct. Anal., (2012). Zbl1257.53055MR2984123DOI10.1007/s00039-012-0176-5
  15. GIGLI, N. - KUWADA, K. - OHTA, S., Heat flow on Alexandrov spaces, To appear on Comm. Pure Appl. Math., (2012). MR3008226DOI10.1002/cpa.21431
  16. GIGLI, N. - OHTA, S.-I., First variation formula in Wasserstein spaces over compact Alexandrov spaces, To appear on Canad. Math. Bull. Zbl1264.53050MR2994677DOI10.4153/CMB-2011-110-3
  17. HEINONEN, J., Nonsmooth calculus, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 163-232. MR2291675DOI10.1090/S0273-0979-07-01140-8
  18. HEINONEN, J. - KOSKELA, P., Quasiconformal maps in metric spaces with controlled geometry, Acta Math., 181 (1998), 1-61. Zbl0915.30018MR1654771DOI10.1007/BF02392747
  19. HEINONEN, J. - KOSKELA, P., A note on Lipschitz functions, upper gradients, and the Poincaré inequality, New Zealand J. Math., 28 (1999), 37-42. Zbl1015.46020MR1691958
  20. KOSKELA, P. - MACMANUS, P., Quasiconformal mappings and Sobolev spaces, Studia Math., 131 (1998), 1-17. Zbl0918.30011MR1628655
  21. KUWADA, K., Duality on gradient estimates and wasserstein controls, Journal of Functional Analysis, 258 (2010), 3758-3774. Zbl1194.53032MR2606871DOI10.1016/j.jfa.2010.01.010
  22. LISINI, S., Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations, 28 (2007), 85-120. Zbl1132.60004MR2267755DOI10.1007/s00526-006-0032-2
  23. LOTT, J. - VILLANI, C., Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991. Zbl1178.53038MR2480619DOI10.4007/annals.2009.169.903
  24. OHTA, S.-I., Finsler interpolation inequalities, Calc. Var. Partial Differential Equations, 36 (2009), 211-249. MR2546027DOI10.1007/s00526-009-0227-4
  25. OHTA, S.-I., Gradient flows on Wasserstein spaces over compact Alexandrov spaces, Amer. J. Math., 131 (2009), 475-516. Zbl1169.53053MR2503990DOI10.1353/ajm.0.0048
  26. OHTA, S.-I. - STURM, K.-T., Heat flow on Finsler manifolds, Comm. Pure Appl. Math., 62 (2009), 1386-1433. Zbl1176.58012MR2547978DOI10.1002/cpa.20273
  27. PETRUNIN, A., Alexandrov meets lott-villani-sturm, arXiv:1003.5948v1, (2010). Zbl1247.53038MR2869253
  28. RAJALA, T., Improved geodesics for the reduced curvature-dimension condition in branching metric spaces, Discrete Contin. Dyn. Syst., (2011). to appear. MR3007737DOI10.3934/dcds.2013.33.3043
  29. SAVARÉ, G. , Gradient flows and evolution variational inequalities in metric spaces, In preparation, (2010). 
  30. SHANMUGALINGAM, N., Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana, 16 (2000), 243-279. Zbl0974.46038MR1809341DOI10.4171/RMI/275
  31. STURM, K.-T., On the geometry of metric measure spaces. I, Acta Math., 196 (2006), 65-131. MR2237206DOI10.1007/s11511-006-0002-8
  32. VILLANI, C., Optimal transport. Old and new, vol. 338 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. Zbl1156.53003MR2459454DOI10.1007/978-3-540-71050-9
  33. ZHANG, H.-C. - ZHU, X.-P., Ricci curvature on Alexandrov spaces and rigidity theorems, Comm. Anal. Geom., 18 (2010), 503-553. Zbl1230.53064MR2747437DOI10.4310/CAG.2010.v18.n3.a4

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