American Institute of Mathematical Sciences

December  2013, 3(4): 467-487. doi: 10.3934/mcrf.2013.3.467

Ricci curvatures in Carnot groups

 1 Université de Nice-Sophia Antipolis, Labo. J.-A. Dieudonné, UMR CNRS 6621, Parc Valrose, 06108 Nice Cedex 02, France

Received  December 2012 Revised  February 2013 Published  September 2013

We study metric contraction properties for metric spaces associated with left-invariant sub-Riemannian metrics on Carnot groups. We show that ideal sub-Riemannian structures on Carnot groups satisfy such properties and give a lower bound of possible curvature exponents in terms of the datas.
Citation: Ludovic Rifford. Ricci curvatures in Carnot groups. Mathematical Control & Related Fields, 2013, 3 (4) : 467-487. doi: 10.3934/mcrf.2013.3.467
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References:
 [1] Paul W. Y. Lee, Chengbo Li, Igor Zelenko. Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303 [2] Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure & Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307 [3] Erlend Grong, Alexander Vasil’ev. Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover. Journal of Geometric Mechanics, 2011, 3 (2) : 225-260. doi: 10.3934/jgm.2011.3.225 [4] Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155 [5] Stefan Sommer, Anne Marie Svane. Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 391-410. doi: 10.3934/jgm.2017015 [6] Yunlong Huang, P. S. Krishnaprasad. Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-26. doi: 10.3934/dcdss.2020072 [7] Ali Maalaoui. A note on commutators of the fractional sub-Laplacian on Carnot groups. Communications on Pure & Applied Analysis, 2019, 18 (1) : 435-453. doi: 10.3934/cpaa.2019022 [8] Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1139-1144. doi: 10.3934/dcds.2011.30.1139 [9] Bang-Xian Han. New characterizations of Ricci curvature on RCD metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4915-4927. doi: 10.3934/dcds.2018214 [10] Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699 [11] Qingsong Gu, Jiaxin Hu, Sze-Man Ngai. Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates. Communications on Pure & Applied Analysis, 2020, 19 (2) : 641-676. doi: 10.3934/cpaa.2020030 [12] Houda Mokrani. Semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1619-1636. doi: 10.3934/cpaa.2009.8.1619 [13] Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia. The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2535-2560. doi: 10.3934/dcds.2014.34.2535 [14] Scott Crass. Solving the heptic by iteration in two dimensions: Geometry and dynamics under Klein's group of order 168. Journal of Modern Dynamics, 2007, 1 (2) : 175-203. doi: 10.3934/jmd.2007.1.175 [15] Tracy L. Payne. The Ricci flow for nilmanifolds. Journal of Modern Dynamics, 2010, 4 (1) : 65-90. doi: 10.3934/jmd.2010.4.65 [16] Fausto Ferrari, Michele Miranda Jr, Diego Pallara, Andrea Pinamonti, Yannick Sire. Fractional Laplacians, perimeters and heat semigroups in Carnot groups. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 477-491. doi: 10.3934/dcdss.2018026 [17] Shuhong Chen, Zhong Tan. Optimal partial regularity results for nonlinear elliptic systems in Carnot groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3391-3405. doi: 10.3934/dcds.2013.33.3391 [18] Jerome A. Goldstein, Ismail Kombe, Abdullah Yener. A unified approach to weighted Hardy type inequalities on Carnot groups. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2009-2021. doi: 10.3934/dcds.2017085 [19] Sheena D. Branton. Sub-actions for young towers. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 541-556. doi: 10.3934/dcds.2008.22.541 [20] Daniel J. Thompson. A criterion for topological entropy to decrease under normalised Ricci flow. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1243-1248. doi: 10.3934/dcds.2011.30.1243

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