# American Institute of Mathematical Sciences

April  2014, 34(4): 1481-1509. doi: 10.3934/dcds.2014.34.1481

## The Abresch-Gromoll inequality in a non-smooth setting

 1 Université de Nice, Mathématiques, Parc Valrose, 06108 Nice, France 2 Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy

Received  September 2012 Revised  November 2012 Published  October 2013

We prove that the Abresch-Gromoll inequality holds on infinitesimally Hilbertian $CD(K,N)$ spaces in the same form as the one available on smooth Riemannian manifolds.
Citation: Nicola Gigli, Sunra Mosconi. The Abresch-Gromoll inequality in a non-smooth setting. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1481-1509. doi: 10.3934/dcds.2014.34.1481
##### References:
 [1] U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature,, J. Amer. Math. Soc., 3 (1990), 355. doi: 10.1090/S0894-0347-1990-1030656-6. Google Scholar [2] L. Ambrosio and N. Gigli, User's guide to optimal transport theory,, to appear in the CIME Lecture Notes in Mathematics, (). Google Scholar [3] L. Ambrosio, N. Gigli, A. Mondino and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure,, accepted in Trans. Amer. Math. Soc., (2012). Google Scholar [4] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Second edition, (2008). Google Scholar [5] ______, Calculus and heat flows in metric measure spaces with Ricci curvature bounded from below,, accepted in Invent. Math., (2013). Google Scholar [6] ______, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces,, accepted in Rev. Mat. Iberoam., (2012). Google Scholar [7] ______, Metric measure spaces with Riemannian Ricci curvature bounded from below,, submitted, (2011). Google Scholar [8] A. Björn and J. Björn, "Nonlinear Potential Theory on Metric Spaces,", EMS Tracts in Mathematics, 17 (2011). doi: 10.4171/099. Google Scholar [9] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces,, Geom. Funct. Anal., 9 (1999), 428. doi: 10.1007/s000390050094. Google Scholar [10] J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products,, Ann. of Math. (2), 144 (1996), 189. doi: 10.2307/2118589. Google Scholar [11] N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability,, Calc. Var. PDE, 39 (2010), 101. doi: 10.1007/s00526-009-0303-9. Google Scholar [12] ______, On the differential structure of metric measure spaces and applications,, accepted in Memoirs of the AMS, (2013). Google Scholar [13] N. Gigli, K. Kuwada and S.-i. Ohta, Heat flow on Alexandrov spaces,, Communications on Pure and Applied Mathematics, 66 (2013), 307. doi: 10.1002/cpa.21431. Google Scholar [14] N. Gigli and A. Mondino, A PDE approach to nonlinear potential theory in metric measure spaces,, Journal de Mathématiques Pures et Appliquées, (2013). doi: 10.1016/j.matpur.2013.01.011. Google Scholar [15] N. Gigli, A. Mondino and G. Savaré, A notion of convergence of non-compact metric measure spaces and applications,, preprint, (2013). Google Scholar [16] J. Lott and C. Villani, Weak curvature bounds and functional inequalities,, J. Funct. Anal., 245 (2007), 311. doi: 10.1016/j.jfa.2006.10.018. Google Scholar [17] ______, Ricci curvature for metric-measure spaces via optimal transport,, Ann. of Math. (2), 169 (2009), 903. doi: 10.4007/annals.2009.169.903. Google Scholar [18] T. Rajala, Local Poincaré inequalities from stable curvature conditions in metric spaces,, Calculus of Variations and Partial Differential Equations, 44 (2012), 477. doi: 10.1007/s00526-011-0442-7. Google Scholar [19] Z. Shen, The non-linear laplacian for Finsler manifolds,, in, 459 (1998), 187. Google Scholar [20] 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 [21] ______, On the geometry of metric measure spaces. II,, Acta Math., 196 (2006), 133. Google Scholar [22] C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften, 338 (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar [23] H. Zhang and X. Zhu, On a new definition of Ricci curvature on Alexandrov spaces,, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 1949. doi: 10.1016/S0252-9602(10)60185-3. Google Scholar

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##### References:
 [1] U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature,, J. Amer. Math. Soc., 3 (1990), 355. doi: 10.1090/S0894-0347-1990-1030656-6. Google Scholar [2] L. Ambrosio and N. Gigli, User's guide to optimal transport theory,, to appear in the CIME Lecture Notes in Mathematics, (). Google Scholar [3] L. Ambrosio, N. Gigli, A. Mondino and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure,, accepted in Trans. Amer. Math. Soc., (2012). Google Scholar [4] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Second edition, (2008). Google Scholar [5] ______, Calculus and heat flows in metric measure spaces with Ricci curvature bounded from below,, accepted in Invent. Math., (2013). Google Scholar [6] ______, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces,, accepted in Rev. Mat. Iberoam., (2012). Google Scholar [7] ______, Metric measure spaces with Riemannian Ricci curvature bounded from below,, submitted, (2011). Google Scholar [8] A. Björn and J. Björn, "Nonlinear Potential Theory on Metric Spaces,", EMS Tracts in Mathematics, 17 (2011). doi: 10.4171/099. Google Scholar [9] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces,, Geom. Funct. Anal., 9 (1999), 428. doi: 10.1007/s000390050094. Google Scholar [10] J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products,, Ann. of Math. (2), 144 (1996), 189. doi: 10.2307/2118589. Google Scholar [11] N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability,, Calc. Var. PDE, 39 (2010), 101. doi: 10.1007/s00526-009-0303-9. Google Scholar [12] ______, On the differential structure of metric measure spaces and applications,, accepted in Memoirs of the AMS, (2013). Google Scholar [13] N. Gigli, K. Kuwada and S.-i. Ohta, Heat flow on Alexandrov spaces,, Communications on Pure and Applied Mathematics, 66 (2013), 307. doi: 10.1002/cpa.21431. Google Scholar [14] N. Gigli and A. Mondino, A PDE approach to nonlinear potential theory in metric measure spaces,, Journal de Mathématiques Pures et Appliquées, (2013). doi: 10.1016/j.matpur.2013.01.011. Google Scholar [15] N. Gigli, A. Mondino and G. Savaré, A notion of convergence of non-compact metric measure spaces and applications,, preprint, (2013). Google Scholar [16] J. Lott and C. Villani, Weak curvature bounds and functional inequalities,, J. Funct. Anal., 245 (2007), 311. doi: 10.1016/j.jfa.2006.10.018. Google Scholar [17] ______, Ricci curvature for metric-measure spaces via optimal transport,, Ann. of Math. (2), 169 (2009), 903. doi: 10.4007/annals.2009.169.903. Google Scholar [18] T. Rajala, Local Poincaré inequalities from stable curvature conditions in metric spaces,, Calculus of Variations and Partial Differential Equations, 44 (2012), 477. doi: 10.1007/s00526-011-0442-7. Google Scholar [19] Z. Shen, The non-linear laplacian for Finsler manifolds,, in, 459 (1998), 187. Google Scholar [20] 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 [21] ______, On the geometry of metric measure spaces. II,, Acta Math., 196 (2006), 133. Google Scholar [22] C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften, 338 (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar [23] H. Zhang and X. Zhu, On a new definition of Ricci curvature on Alexandrov spaces,, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 1949. doi: 10.1016/S0252-9602(10)60185-3. Google Scholar
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