# American Institute of Mathematical Sciences

July  2013, 33(7): 3043-3056. doi: 10.3934/dcds.2013.33.3043

## Improved geodesics for the reduced curvature-dimension condition in branching metric spaces

 1 Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56127 Pisa, Italy

Received  March 2012 Revised  March 2012 Published  January 2013

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition $CD^*(K,N)$ we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of $CD^*(K,N)$ also for intermediate times and in addition the measures along these geodesics have an upper-bound on their densities. This upper-bound depends on the bounds for the densities of the end-point measures, the lower-bound $K$ for the Ricci-curvature, the upper-bound $N$ for the dimension, and on the diameter of the union of the supports of the end-point measures.
Citation: Tapio Rajala. Improved geodesics for the reduced curvature-dimension condition in branching metric spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3043-3056. doi: 10.3934/dcds.2013.33.3043
##### References:
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##### References:
 [1] K. Bacher and K.-T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces,, Journal Funct. Anal., 259 (2010), 28. doi: 10.1016/j.jfa.2010.03.024. Google Scholar [2] F. Cavalletti and K.-T. Sturm, Local curvature-dimension condition implies measure-contraction property,, Journal Funct. Anal., 262 (2012), 5110. doi: 10.1016/j.jfa.2012.02.015. Google Scholar [3] Q. Deng and K.-T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces II,, Journal Funct. Anal., 260 (2011), 3718. doi: 10.1016/j.jfa.2011.02.026. Google Scholar [4] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport,, Ann. of Math., 169 (2009), 903. doi: 10.4007/annals.2009.169.903. Google Scholar [5] T. Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm,, Journal Funct. Anal., 263 (2012), 896. doi: 10.1016/j.jfa.2012.05.006. Google Scholar [6] T. Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces,, Calc. Var. Partial Differential Equations, 44 (2012), 477. doi: 10.1007/s00526-011-0442-7. Google Scholar [7] K.-T. Sturm, On the geometry of metric measure spaces. I,, Acta Math., 196 (2006), 65. doi: 10.1007/s11511-006-0002-8. Google Scholar [8] K.-T. Sturm, On the geometry of metric measure spaces. II,, Acta Math., 196 (2006), 133. doi: 10.1007/s11511-006-0003-7. Google Scholar [9] C. Villani, "Optimal Transport. Old and New,", 338 of Grundlehren der Mathematischen Wissenschaften, 338 (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar
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