# 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 and Continuous Dynamical Systems, 2013, 33 (7) : 3043-3056. doi: 10.3934/dcds.2013.33.3043
##### 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), 28-56. doi: 10.1016/j.jfa.2010.03.024. [2] F. Cavalletti and K.-T. Sturm, Local curvature-dimension condition implies measure-contraction property, Journal Funct. Anal., 262 (2012), 5110-5127. doi: 10.1016/j.jfa.2012.02.015. [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), 3718-3725. doi: 10.1016/j.jfa.2011.02.026. [4] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math., 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903. [5] T. Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm, Journal Funct. Anal., 263 (2012), 896-924. doi: 10.1016/j.jfa.2012.05.006. [6] T. Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations, 44 (2012), 477-494. doi: 10.1007/s00526-011-0442-7. [7] K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8. [8] K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math., 196 (2006), 133-177. doi: 10.1007/s11511-006-0003-7. [9] C. Villani, "Optimal Transport. Old and New," 338 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

show all references

##### 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), 28-56. doi: 10.1016/j.jfa.2010.03.024. [2] F. Cavalletti and K.-T. Sturm, Local curvature-dimension condition implies measure-contraction property, Journal Funct. Anal., 262 (2012), 5110-5127. doi: 10.1016/j.jfa.2012.02.015. [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), 3718-3725. doi: 10.1016/j.jfa.2011.02.026. [4] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math., 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903. [5] T. Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm, Journal Funct. Anal., 263 (2012), 896-924. doi: 10.1016/j.jfa.2012.05.006. [6] T. Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations, 44 (2012), 477-494. doi: 10.1007/s00526-011-0442-7. [7] K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8. [8] K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math., 196 (2006), 133-177. doi: 10.1007/s11511-006-0003-7. [9] C. Villani, "Optimal Transport. Old and New," 338 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.
 [1] Bang-Xian Han. New characterizations of Ricci curvature on RCD metric measure spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4915-4927. doi: 10.3934/dcds.2018214 [2] Ugo Bessi. The stochastic value function in metric measure spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076 [3] Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773 [4] Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327 [5] Giuseppe Savaré. Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641 [6] Alexander J. Zaslavski. Stability of a turnpike phenomenon for a class of optimal control systems in metric spaces. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 245-260. doi: 10.3934/naco.2011.1.245 [7] Jaeyoo Choy, Hahng-Yun Chu. On the dynamics of flows on compact metric spaces. Communications on Pure and Applied Analysis, 2010, 9 (1) : 103-108. doi: 10.3934/cpaa.2010.9.103 [8] Rinaldo M. Colombo, Graziano Guerra. Differential equations in metric spaces with applications. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 733-753. doi: 10.3934/dcds.2009.23.733 [9] Saul Mendoza-Palacios, Onésimo Hernández-Lerma. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics and Games, 2017, 4 (4) : 319-333. doi: 10.3934/jdg.2017017 [10] Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor. Soliton solutions for the elastic metric on spaces of curves. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1161-1185. doi: 10.3934/dcds.2018049 [11] Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 629-654. doi: 10.3934/dcdsb.2010.14.629 [12] Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016 [13] Byung-Soo Lee. Existence and convergence results for best proximity points in cone metric spaces. Numerical Algebra, Control and Optimization, 2014, 4 (2) : 133-140. doi: 10.3934/naco.2014.4.133 [14] Roberta Ghezzi, Frédéric Jean. A new class of $(H^k,1)$-rectifiable subsets of metric spaces. Communications on Pure and Applied Analysis, 2013, 12 (2) : 881-898. doi: 10.3934/cpaa.2013.12.881 [15] Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629 [16] Alexander Mielke, Riccarda Rossi, Giuseppe Savaré. Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 585-615. doi: 10.3934/dcds.2009.25.585 [17] Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427 [18] Ryan Alvarado, Irina Mitrea, Marius Mitrea. Whitney-type extensions in quasi-metric spaces. Communications on Pure and Applied Analysis, 2013, 12 (1) : 59-88. doi: 10.3934/cpaa.2013.12.59 [19] Moisey Guysinsky, Serge Yaskolko. Coincidence of various dimensions associated with metrics and measures on metric spaces. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 591-603. doi: 10.3934/dcds.1997.3.591 [20] Adrian Petruşel, Radu Precup, Marcel-Adrian Şerban. On the approximation of fixed points for non-self mappings on metric spaces. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 733-747. doi: 10.3934/dcdsb.2019264

2020 Impact Factor: 1.392