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Emergent dynamics of the Kuramoto ensemble under the effect of inertia
New characterizations of Ricci curvature on RCD metric measure spaces
Institute for applied mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany |
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.
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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