# 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 and Continuous Dynamical Systems, 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), 355-374. doi: 10.1090/S0894-0347-1990-1030656-6. [2] L. Ambrosio and N. Gigli, User's guide to optimal transport theory,, to appear in the CIME Lecture Notes in Mathematics, (). [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., arXiv:1207.4924 (2012). [4] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [5] ______, Calculus and heat flows in metric measure spaces with Ricci curvature bounded from below, accepted in Invent. Math., arXiv:1106.2090, (2013). [6] ______, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, accepted in Rev. Mat. Iberoam., arXiv:1111.3730 (2012). [7] ______, Metric measure spaces with Riemannian Ricci curvature bounded from below, submitted, arXiv:1109.0222, (2011). [8] A. Björn and J. Björn, "Nonlinear Potential Theory on Metric Spaces," EMS Tracts in Mathematics, 17, European Mathematical Society (EMS), Zürich, 2011. doi: 10.4171/099. [9] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517. doi: 10.1007/s000390050094. [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), 189-237. doi: 10.2307/2118589. [11] N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calc. Var. PDE, 39 (2010), 101-120. doi: 10.1007/s00526-009-0303-9. [12] ______, On the differential structure of metric measure spaces and applications, accepted in Memoirs of the AMS, arXiv:1205.6622 (2013). [13] N. Gigli, K. Kuwada and S.-i. Ohta, Heat flow on Alexandrov spaces, Communications on Pure and Applied Mathematics, 66 (2013), 307-331. doi: 10.1002/cpa.21431. [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. [15] N. Gigli, A. Mondino and G. Savaré, A notion of convergence of non-compact metric measure spaces and applications, preprint, (2013). [16] J. Lott and C. Villani, Weak curvature bounds and functional inequalities, J. Funct. Anal., 245 (2007), 311-333. doi: 10.1016/j.jfa.2006.10.018. [17] ______, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903. [18] T. Rajala, Local Poincaré inequalities from stable curvature conditions in metric spaces, Calculus of Variations and Partial Differential Equations, 44 (2012), 477-494. doi: 10.1007/s00526-011-0442-7. [19] Z. Shen, The non-linear laplacian for Finsler manifolds, in "The Theory of Finslerian Laplacians and Applications," Math. Appl., 459, Kluwer Acad. Publ., Dordrecht, (1998), 187-198. [20] K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8. [21] ______, On the geometry of metric measure spaces. II, Acta Math., 196 (2006), 133-177. [22] C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. [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), 1949-1974. doi: 10.1016/S0252-9602(10)60185-3.

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##### References:
 [1] U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc., 3 (1990), 355-374. doi: 10.1090/S0894-0347-1990-1030656-6. [2] L. Ambrosio and N. Gigli, User's guide to optimal transport theory,, to appear in the CIME Lecture Notes in Mathematics, (). [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., arXiv:1207.4924 (2012). [4] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [5] ______, Calculus and heat flows in metric measure spaces with Ricci curvature bounded from below, accepted in Invent. Math., arXiv:1106.2090, (2013). [6] ______, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, accepted in Rev. Mat. Iberoam., arXiv:1111.3730 (2012). [7] ______, Metric measure spaces with Riemannian Ricci curvature bounded from below, submitted, arXiv:1109.0222, (2011). [8] A. Björn and J. Björn, "Nonlinear Potential Theory on Metric Spaces," EMS Tracts in Mathematics, 17, European Mathematical Society (EMS), Zürich, 2011. doi: 10.4171/099. [9] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517. doi: 10.1007/s000390050094. [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), 189-237. doi: 10.2307/2118589. [11] N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calc. Var. PDE, 39 (2010), 101-120. doi: 10.1007/s00526-009-0303-9. [12] ______, On the differential structure of metric measure spaces and applications, accepted in Memoirs of the AMS, arXiv:1205.6622 (2013). [13] N. Gigli, K. Kuwada and S.-i. Ohta, Heat flow on Alexandrov spaces, Communications on Pure and Applied Mathematics, 66 (2013), 307-331. doi: 10.1002/cpa.21431. [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. [15] N. Gigli, A. Mondino and G. Savaré, A notion of convergence of non-compact metric measure spaces and applications, preprint, (2013). [16] J. Lott and C. Villani, Weak curvature bounds and functional inequalities, J. Funct. Anal., 245 (2007), 311-333. doi: 10.1016/j.jfa.2006.10.018. [17] ______, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903. [18] T. Rajala, Local Poincaré inequalities from stable curvature conditions in metric spaces, Calculus of Variations and Partial Differential Equations, 44 (2012), 477-494. doi: 10.1007/s00526-011-0442-7. [19] Z. Shen, The non-linear laplacian for Finsler manifolds, in "The Theory of Finslerian Laplacians and Applications," Math. Appl., 459, Kluwer Acad. Publ., Dordrecht, (1998), 187-198. [20] K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8. [21] ______, On the geometry of metric measure spaces. II, Acta Math., 196 (2006), 133-177. [22] C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. [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), 1949-1974. doi: 10.1016/S0252-9602(10)60185-3.
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