New characterizations of Ricci curvature on RCD metric measure spaces

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October 2018, 38(10): 4915-4927. doi: 10.3934/dcds.2018214

New characterizations of Ricci curvature on RCD metric measure spaces

Bang-Xian Han ^,

Institute for applied mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

* Corresponding author: Bang-Xian Han

Received September 2017 Revised April 2018 Published July 2018

We prove that on a large family of metric measure spaces, if the $L^p$-gradient estimate for heat flows holds for some $p>2$, then the $L^1$-gradient estimate also holds. This result extends Savaré's result on metric measure spaces, and provides a new proof to von Renesse-Sturm theorem on smooth metric measure spaces. As a consequence, we propose a new analysis object based on Gigli's measure-valued Ricci tensor, to characterize the Ricci curvature of RCD space in a local way. In the proof we adopt an iteration technique based on non-smooth Bakry-Émery theory, which is a new method to study the curvature dimension condition of metric measure spaces.

Keywords: Bakry-Émery theory, curvature dimension condition, gradient estimate, heat flow, metric measure space, Ricci curvature.

Mathematics Subject Classification: Primary: 47D07, 30L99; Secondary: 51F99.

Citation: 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

References:

[1]	L. Ambrosio, N. Gigli, A. Mondino and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with $σ$-finite measure, Trans. Amer. Math. Soc., 367 (2015), 4661-4701. doi: 10.1090/S0002-9947-2015-06111-X.
[2]	L. Ambrosio, N. Gigli and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., 195 (2014), 289-391. doi: 10.1007/s00222-013-0456-1.
[3]	L. Ambrosio, N. Gigli and G. Savaré, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 969-996. doi: 10.4171/RMI/746.
[4]	L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163 (2014), 1405-1490. doi: 10.1215/00127094-2681605.
[5]	L. Ambrosio, N. Gigli and G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab., 43 (2015), 339-404. doi: 10.1214/14-AOP907.
[6]	L. Ambrosio, A. Mondino and G. Savaré, On the Bakry-Émery condition, the gradient estimates and the local-to-global property of $\text{RC}{\text{ D}^{}}\left( K,N \right)$ metric measure spaces, J. Geom. Anal.*, 26 (2016), 24-56. doi: 10.1007/s12220-014-9537-7.
[7]	D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes, in Lectures on probability theory (Saint-Flour, 1992), vol. 1581 of Lecture Notes in Math. Springer, Berlin, 1994, pp. 1-114. doi: 10.1007/BFb0073872.
[8]	N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, De Gruyter Studies in Mathematics, 14. Walter de Gruyter & Co., Berlin, 1991. doi: 10.1515/9783110858389.
[9]	Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, vol. 35 of London Mathematical Society Monographs Series, Princeton University Press, Princeton, NJ, 2012.
[10]	N. Gigli, Nonsmooth differential geometry-approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc., 251 (2018), ⅵ+161pp.
[11]	N. Gigli, On the differential structure of metric measure spaces and applications Mem. Amer. Math. Soc., 236 (2015), ⅵ+91pp. doi: 10.1090/memo/1113.
[12]	B.-X. Han, Ricci tensor on RCD^(K, N) spaces, J. Geom. Anal.*, 28 (2018), 1295-1314. doi: 10.1007/s12220-017-9863-7.
[13]	J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903.
[14]	M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940. doi: 10.1002/cpa.20060.
[15]	G. Savaré, Self-improvement of the Bakry-émery condition and Wasserstein contraction of the heat flow in ${RCD(K, ∞)}$ metric measure spaces, Disc. Cont. Dyn. Sist. A, 34 (2014), 1641-1661. doi: 10.3934/dcds.2014.34.1641.
[16]	K.-T. Sturm, On the geometry of metric measure spaces Ⅰ, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8.
[17]	K.-T. Sturm, Ricci tensor for diffusion operators and curvature-dimension inequalities under conformal transformations and time changes, J. Funct. Anal., 275 (2018), 793-829. doi: 10.1016/j.jfa.2018.03.022.
[18]	C. Villani, Optimal Transport. Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

show all references

References:

[1]	L. Ambrosio, N. Gigli, A. Mondino and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with $σ$-finite measure, Trans. Amer. Math. Soc., 367 (2015), 4661-4701. doi: 10.1090/S0002-9947-2015-06111-X.
[2]	L. Ambrosio, N. Gigli and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., 195 (2014), 289-391. doi: 10.1007/s00222-013-0456-1.
[3]	L. Ambrosio, N. Gigli and G. Savaré, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 969-996. doi: 10.4171/RMI/746.
[4]	L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163 (2014), 1405-1490. doi: 10.1215/00127094-2681605.
[5]	L. Ambrosio, N. Gigli and G. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab., 43 (2015), 339-404. doi: 10.1214/14-AOP907.
[6]	L. Ambrosio, A. Mondino and G. Savaré, On the Bakry-Émery condition, the gradient estimates and the local-to-global property of $\text{RC}{\text{ D}^{}}\left( K,N \right)$ metric measure spaces, J. Geom. Anal.*, 26 (2016), 24-56. doi: 10.1007/s12220-014-9537-7.
[7]	D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes, in Lectures on probability theory (Saint-Flour, 1992), vol. 1581 of Lecture Notes in Math. Springer, Berlin, 1994, pp. 1-114. doi: 10.1007/BFb0073872.
[8]	N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, De Gruyter Studies in Mathematics, 14. Walter de Gruyter & Co., Berlin, 1991. doi: 10.1515/9783110858389.
[9]	Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, vol. 35 of London Mathematical Society Monographs Series, Princeton University Press, Princeton, NJ, 2012.
[10]	N. Gigli, Nonsmooth differential geometry-approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc., 251 (2018), ⅵ+161pp.
[11]	N. Gigli, On the differential structure of metric measure spaces and applications Mem. Amer. Math. Soc., 236 (2015), ⅵ+91pp. doi: 10.1090/memo/1113.
[12]	B.-X. Han, Ricci tensor on RCD^(K, N) spaces, J. Geom. Anal.*, 28 (2018), 1295-1314. doi: 10.1007/s12220-017-9863-7.
[13]	J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903.
[14]	M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940. doi: 10.1002/cpa.20060.
[15]	G. Savaré, Self-improvement of the Bakry-émery condition and Wasserstein contraction of the heat flow in ${RCD(K, ∞)}$ metric measure spaces, Disc. Cont. Dyn. Sist. A, 34 (2014), 1641-1661. doi: 10.3934/dcds.2014.34.1641.
[16]	K.-T. Sturm, On the geometry of metric measure spaces Ⅰ, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8.
[17]	K.-T. Sturm, Ricci tensor for diffusion operators and curvature-dimension inequalities under conformal transformations and time changes, J. Funct. Anal., 275 (2018), 793-829. doi: 10.1016/j.jfa.2018.03.022.
[18]	C. Villani, Optimal Transport. Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

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