A rough curvature-dimension condition for metric measure spaces

 Access to full text

 Full (PDF)

1. [1] Aleksandrov A.D., A theorem on triangles in a metric space and some of its applications, Trudy Mat. Inst. Steklov., 1951, 38, 5–23 (in Russian; translated into German and combined with more material in [2]) 
2. [2] Alexandrow A.D., Über eine Verallgemeinerung der Riemannschen Geometrie, Schr. Forschungsinst. Math., 1957, 1, 33–84 Zbl0077.35702
3. [3] Bakry D., Émery M., Diffusions hypercontractives, In: Séminaire de Probabilités, XIX, 1983/84, Lecture Notes in Math., 1123, Springer, Berlin, 1985, 177–206 http://dx.doi.org/10.1007/BFb0075847 
4. [4] Bonciocat A.-I., Sturm K.-T., Mass transportation and rough curvature bounds for discrete spaces, J. Funct. Anal., 2009, 256(9), 2944–2966 http://dx.doi.org/10.1016/j.jfa.2009.01.029 Zbl1184.28015
5. [5] Bonnefont M., A discrete version of the Brunn-Minkowski inequality and its stability, Ann. Math. Blaise Pascal, 2009, 16(2), 245–257 http://dx.doi.org/10.5802/ambp.264 Zbl1197.52003
6. [6] Dudley R.M, Real Analysis and Probability, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, 1989 
7. [7] Erbar M., Maas J., Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal., 2012, 206(3), 997–1038 http://dx.doi.org/10.1007/s00205-012-0554-z Zbl1256.53028
8. [8] Forman R., Bochner’s method for cell complexes and combinatorial Ricci curvature, Discrete Comput. Geom., 2003, 29(3), 323–374 http://dx.doi.org/10.1007/s00454-002-0743-x Zbl1040.53040
9. [9] Fukaya K., Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math., 1987, 87(3), 517–547 http://dx.doi.org/10.1007/BF01389241 Zbl0589.58034
10. [10] Gromov M., Hyperbolic groups, In: Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75–263 http://dx.doi.org/10.1007/978-1-4613-9586-7_3 
11. [11] Gromov M., Metric Structures for Riemannian and Non-Riemannian Spaces, Progr. Math., 152, Birkhäuser, Boston, 1999 
12. [12] Higuchi Y., Combinatorial curvature for planar graphs, J. Graph Theory, 2001, 38(4), 220–229 http://dx.doi.org/10.1002/jgt.10004 Zbl0996.05041
13. [13] Ishida M., Pseudo-curvature of a graph, Workshop on Topological Graph Theory, Yokohama National University, 1990 (lecture) 
14. [14] Lin Y., Yau S.-T., Ricci curvature and eigenvalue estimate on locally finite graphs, Math. Res. Lett., 2010, 17(2), 343–356 http://dx.doi.org/10.4310/MRL.2010.v17.n2.a13 Zbl1232.31003
15. [15] Lott J., Villani C., Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math., 2009, 169(3), 903–991 http://dx.doi.org/10.4007/annals.2009.169.903 Zbl1178.53038
16. [16] Maas J., Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 2011, 261(8), 2250–2292 http://dx.doi.org/10.1016/j.jfa.2011.06.009 Zbl1237.60058
17. [17] Mielke A., Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations, 2013, 48(1–2), 1–31 http://dx.doi.org/10.1007/s00526-012-0538-8 Zbl1282.60072
18. [18] Ollivier Y., Ricci curvature of Markov chains on metric spaces, J. Funct. Anal., 2009, 256(3), 810–864 http://dx.doi.org/10.1016/j.jfa.2008.11.001 Zbl1181.53015
19. [19] Roe J., Lectures on Coarse Geometry, Univ. Lecture Ser., 31, American Mathematical Society, Providence, 2003 Zbl1042.53027
20. [20] Sturm K.-T., On the geometry of metric measure spaces I, Acta Math., 2006, 196(1), 65–131 http://dx.doi.org/10.1007/s11511-006-0002-8 
21. [21] Sturm K.-T., On the geometry of metric measure spaces II, Acta Math., 2006, 196(1), 133–177 http://dx.doi.org/10.1007/s11511-006-0003-7 Zbl1106.53032
22. [22] Villani C., Topics in Optimal Transportation, Grad. Stud. Math., 58, American Mathematical Society, Providence, 2003 Zbl1106.90001
23. [23] Villani C., Optimal Transport, Grundlehren Math. Wiss., 338, Springer, Berlin, 2009 http://dx.doi.org/10.1007/978-3-540-71050-9