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On quotients of spaces with Ricci curvature bounded below. (English) Zbl 1396.58013

Let \((M, g)\) be a smooth Riemannian manifold and \(G\) a compact Lie group acting on \(M\) effectively and by isometries. A lower bound of the sectional curvature of \((M, g)\) is also a bound for the curvature of the quotient space. The authors of the paper prove some stability properties for synthetic Ricci curvature lower bounds. Precisely, they show that such stability holds for quotients of \(RCD^*(K, N)\)-spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. They approach the same problem for the \(CD/CD^*\) and \(MCP\) curvature-dimension conditions. They provide geometric applications which include: a generalization of Kobayashi’s Classification Theorem of homogeneous manifolds to \(RCD^*(K, N)\)-spaces with essential minimal dimension \(n\leq N\); a structure theorem for \(RCD^*(K, N)\)-spaces admitting actions by large (compact) groups; and geometric rigidity results for orbifolds such as Cheng’s maximal diameter and maximal volume rigidity theorems. They apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of (super-)Ricci flows.

MSC:

58D19 Group actions and symmetry properties
57S15 Compact Lie groups of differentiable transformations
53C20 Global Riemannian geometry, including pinching
53C24 Rigidity results
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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References:

[1] Ambrosio, L.; Colombo, M.; Di Marino, S., Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope, (Variational Methods for Evolving Objects, Adv. Stud. Pure Math., vol. 67, (2015), Math. Soc. Japan Tokyo), 1-58 · Zbl 1370.46018
[2] Ambrosio, L.; Gigli, N., User’s guide to optimal transport theory, modeling and optimization of flows on networks, (Lecture Notes in Mathematics, (2013), Springer Berlin), 1-155
[3] Ambrosio, L.; Gigli, N.; Savaré, G., Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Rev. Mat. Iberoam., 29, 3, 969-996, (2013) · Zbl 1287.46027
[4] Ambrosio, L.; Gigli, N.; Savaré, G., Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., 195, 2, 289-391, (2014) · Zbl 1312.53056
[5] Ambrosio, L.; Gigli, N.; Savaré, G., Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163, 1405-1490, (2014) · Zbl 1304.35310
[6] Ambrosio, L.; Gigli, N.; Mondino, A.; Rajala, T., Riemannian Ricci curvature lower bounds in metric spaces with σ-finite measure, Trans. Amer. Math. Soc., 367, 7, 4661-4701, (2015) · Zbl 1317.53060
[7] Ambrosio, L.; Mondino, A.; Savaré, G., On the Bakry-émery condition, the gradient estimates and the local-to-global property of \(\mathsf{RCD}^\ast(K, N)\) metric measure spaces, J. Geom. Anal., 26, 1, 24-56, (2016) · Zbl 1335.35088
[8] Ambrosio, L.; Mondino, A.; Savaré, G., Nonlinear diffusion equations and curvature conditions in metric measure spaces, Mem. Amer. Math. Soc., (2018), in press
[9] Bacher, K.; Sturm, K.-T., Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal., 259, 28-56, (2010) · Zbl 1196.53027
[10] Bakry, D.; Emery, M., Diffusions hypercontractives, (Sem. de Prob. XIX, Lecture Notes in Math., vol. 1123, (1985), Springer-Verlag), 177-206 · Zbl 0561.60080
[11] Bauer, F.; Horn, P.; Lin, Y.; Lippner, G.; Mangoubi, D.; Yau, S. T., Li-Yau inequality on graphs, J. Differential Geom., 99, 3, 359-405, (2015) · Zbl 1323.35189
[12] Benamou, J. D.; Brenier, Y., A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84, 375-393, (2000) · Zbl 0968.76069
[13] Berestovskii, V.-N., Homogeneous manifolds with an intrinsic metric. I, Sib. Math. J., 29, 6, 887-897, (1989) · Zbl 0686.53041
[14] Berestovskii, V.-N., Homogeneous manifolds with an intrinsic metric. II, Sib. Math. J., 30, 2, 180-191, (1989) · Zbl 0681.53029
[15] Borzellino, J. E., Riemannian geometry of orbifolds, (1992), University of California Los Angeles, Thesis (Ph.D.)
[16] Borzellino, J. E., Orbifolds of maximal diameter, Indiana Univ. Math. J., 42, 1, 37-53, (1993) · Zbl 0801.53031
[17] Burago, Yu.; Gromov, M.; Perelman, G., AD Alexandrov spaces with curvature bounded below, Russian Math. Surveys, 47, 2, 1-58, (1992) · Zbl 0802.53018
[18] Cavalletti, F.; Huesmann, M., Existence and uniqueness of optimal transport maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32, 6, 1367-1377, (2015) · Zbl 1331.49063
[19] Cavalletti, F.; Milman, E., The globalization theorem for the curvature dimension condition, preprint
[20] Cavalletti, F.; Mondino, A., Optimal maps in essentially non-branching spaces, Commun. Contemp. Math., 19, 6, (2017), 27 pp · Zbl 1376.53064
[21] Cheeger, J., Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9, 3, 428-517, (1999) · Zbl 0942.58018
[22] Cheeger, G.; Colding, T. H., On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., 45, 406-480, (1997) · Zbl 0902.53034
[23] Cheeger, J.; Kleiner, B.; Schioppa, A., Infinitesimal structure of differentiability spaces, and metric differentiation, Anal. Geom. Metric Spaces, 4, 1, 104-159, (2016) · Zbl 1360.30047
[24] Chen, B. L.; Zhu, X. P., Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom., 74, 119-154, (2006) · Zbl 1104.53032
[25] Di Marino, S.; Speight, G., The p-weak gradient depends on p, Proc. Amer. Math. Soc., 143, 5239-5252, (2015) · Zbl 1350.46032
[26] Erbar, M.; Kuwada, K.; Sturm, K.-T., On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math., 201, 993-1071, (2015) · Zbl 1329.53059
[27] Erbar, M.; Maas, J., Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal., 206, 3, 997-1038, (2012) · Zbl 1256.53028
[28] Fremlin, D. H., Measure theory, vol. 4, (2002), Torres Fremlin · Zbl 1166.28001
[29] Galaz-Garcia, F.; Guijarro, L., Isometry groups of Alexandrov spaces, Bull. Lond. Math. Soc., 45, 3, 567-579, (2013) · Zbl 1277.53069
[30] Galaz-Garcia, F.; Searle, C., Cohomogeneity one Alexandrov spaces, Transform. Groups, 16, 91-107, (2011) · Zbl 1220.53085
[31] Gigli, N., On the differential structure of metric measure spaces and applications, Memoirs of the AMS, vol. 236(1113), (2015) · Zbl 1325.53054
[32] Gigli, N.; de Philippis, G., From volume cone to metric cone in the nonsmooth setting, Geom. Funct. Anal., 26, 1526-1587, (2016) · Zbl 1356.53049
[33] Gigli, N.; Mondino, A.; Savaré, G., Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc., 111, 5, 1071-1129, (2015) · Zbl 1398.53044
[34] Gigli, N.; Rajala, T.; Sturm, K. T., Optimal maps and exponentiation on finite-dimensional spaces with Ricci curvature bounded from below, J. Geom. Anal., 26, 4, 2914-2929, (2016) · Zbl 1361.53036
[35] Grove, K., Geometry of, and via, symmetries, (Conformal, Riemannian and Lagrangian Geometry, Univ. Lecture Ser., vol. 27, (2002)), 31-53
[36] Guijarro, L.; Santos-Rodríguez, J., On the isometry group of \(\mathsf{RCD}^\ast(K, N)\)-spaces, Manuscripta Math., (2018)
[37] Hamilton, R., Three-manifolds with positive Ricci curvature, J. Differential Geom., 17, 255-306, (1982) · Zbl 0504.53034
[38] Ishihara, S., Homogeneous Riemannian spaces of four dimensions, J. Math. Soc. Japan, 7, 345-370, (1955) · Zbl 0067.39602
[39] Keith, S., A differentiable structure for metric measure spaces, Adv. Math., 183, 2, 271-315, (2004) · Zbl 1077.46027
[40] Kell, M., q-heat flow and the gradient flow of the renyi entropy in the p-Wasserstein space, J. Funct. Anal., 271, 8, 2045-2089, (2016) · Zbl 1375.53056
[41] Kell, M., On interpolation and curvature via Wasserstein geodesics, Adv. Calc. Var., 10, 2, 125-167, (2017) · Zbl 1454.58017
[42] Kell, M., On Cheeger and Sobolev differentials in metric measure spaces, preprint
[43] Ketterer, C., Cones over metric measure spaces and the maximal diameter theorem, J. Math. Pures Appl., 103, 5, 1228-1275, (2015) · Zbl 1317.53064
[44] Ketterer, C.; Mondino, A., Sectional and intermediate Ricci curvature lower bounds via optimal transport, Adv. Math., 329, 781-818, (2018) · Zbl 1387.53027
[45] Ketterer, C.; Rajala, T., Failure of topological rigidity results for the measure contraction property, Potential Anal., 42, 3, 645-655, (2015) · Zbl 1321.53045
[46] Kitabeppu, Y.; Lakzian, S., Characterization of low dimensional \(\mathsf{RCD}^\ast(K, N)\) spaces, Anal. Geom. Metric Spaces, 4, 187-215, (2016) · Zbl 1348.53046
[47] Kleiner, B.; Lott, J., Geometrization of three-dimensional orbifolds via Ricci flow, Astérisque, 365, 101-177, (2014)
[48] Kobayashi, S., Transformation groups in differential geometry, Classics Math., (1995), Springer-Verlag Berlin, Reprint of the 1972 edition · Zbl 0829.53023
[49] Kobayashi, S.; Nagano, T., Riemannian manifolds with abundant isometries, (Differential Geometry (in Honor of Kentaro Yano), (1972), Kinokuniya Tokyo), 195-219 · Zbl 0251.53036
[50] Kotschwar, B. L., Backwards uniqueness for the Ricci flow, Int. Math. Res. Not., 2010, 21, 4064-4097, (2010) · Zbl 1211.53086
[51] Lin, Y.; Yau, S. T., Ricci curvature and eigenvalue estimate on locally finite graphs, Math. Res. Lett., 17, 343-356, (2010) · Zbl 1232.31003
[52] Lott, J., Some geometric properties of the Bakry-émery Ricci tensor, Comment. Math. Helv., 78, 865-883, (2003) · Zbl 1038.53041
[53] Lott, J.; Villani, C., Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169, 903-991, (2009) · Zbl 1178.53038
[54] Maas, J., Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261, 8, 2250-2292, (2011) · Zbl 1237.60058
[55] Mielke, A., Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations, 48, 1, 1-31, (2013) · Zbl 1282.60072
[56] Mondello, I., An obata singular theorem for stratified spaces, Trans. Amer. Math. Soc., 370, 6, 4147-4175, (2018) · Zbl 1387.53046
[57] Mondino, A.; Naber, A., Structure theory of metric-measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc., (2018), in press
[58] Mugnolo, D., Parabolic theory of the discrete p-Laplace operator, Nonlinear Anal., 87, 33-60, (2013) · Zbl 1285.39004
[59] Ohta, S., Finsler interpolation inequalities, Calc. Var. Partial Differential Equations, 36, 2, 211-249, (2009) · Zbl 1175.49044
[60] Ollivier, Y., Ricci curvature of Markov chains on metric spaces, J. Funct. Anal., 256, 810-864, (2009) · Zbl 1181.53015
[61] O’Neill, B., The fundamental equations of a submersion, Michigan Math. J., 13, 459-469, (1966) · Zbl 0145.18602
[62] Pansu, P., Métriques de Carnot-caratheodory et quasiisométries des espaces symmétriques de rang un, Ann. Math., 129, 1-60, (1989) · Zbl 0678.53042
[63] Pro, C.; Wilhelm, F., Riemannian submersions need not preserve positive Ricci curvature, Proc. Amer. Math. Soc., 142, 7, 2529-2535, (2014) · Zbl 1293.53045
[64] Rajala, T., Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations, 44, 3, 477-494, (2012) · Zbl 1250.53040
[65] Rajala, T.; Sturm, K. T., Non-branching geodesics and optimal maps in strong \(\mathsf{CD}(K, \infty)\)-spaces, Calc. Var. Partial Differential Equations, 50, 831-846, (2014) · Zbl 1296.53088
[66] Schioppa, A., Derivations and alberti representations, Adv. Math., 293, 436-528, (2016) · Zbl 1335.53053
[67] Semmes, S., Bilipschitz embeddings of metric spaces into Euclidean spaces, Publ. Mat., 43, 2, 571-653, (1999) · Zbl 1131.30337
[68] Sosa, G., The isometry group of an \(\mathsf{RCD}^\ast\)-space is Lie, Potential Anal., (2017)
[69] Sturm, K. T., On the geometry of metric measure spaces. I, Acta Math., 196, 65-131, (2006) · Zbl 1105.53035
[70] Sturm, K. T., On the geometry of metric measure spaces. II, Acta Math., 196, 133-177, (2006) · Zbl 1106.53032
[71] Sturm, K. T., Super-Ricci flows for metric measure spaces. I, preprint
[72] van Limbeek, W., Isometry types of frame bundles, Pacific J. Math., 285, 2, 393-426, (2016) · Zbl 1360.53047
[73] Villani, C., Optimal transport. old and new, Grundlehren der Mathematischen Wissenschaften, vol. 338, (2009), Springer-Verlag Berlin · Zbl 1156.53003
[74] Walschap, G., Metric foliations and curvature, J. Geom. Anal., 2, 4, 373-381, (1992) · Zbl 0769.53021
[75] Ziller, W., On the geometry of cohomogeneity one manifolds with positive curvature, Riemannian topology and geometric structures on manifolds, Progr. Math., 271, 233-262, (2009) · Zbl 1180.53037
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