Abstract
Since non-compact \(\text {RCD}(0, N)\) spaces have at least linear volume growth, we study non-compact \(\text {RCD}(0,N)\) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grows at most linearly on a non-compact \(\text {RCD}(0,N)\) space satisfying the linear volume growth condition. Another main result in this paper is a rigidity theorem at the non-compact end for a \(\text {RCD}(0,N)\) space with strongly minimal volume growth. These results generalize some theorems on non-compact manifolds with non-negative Ricci curvature to non-smooth settings.
Similar content being viewed by others
References
Ambrosio, L., Gigli, N.: Lecture Notes in Mathematics. A User’s Guide to Optimal Transport. Modelling and Optimisation of Flows on Networks. Springer, Heidelberg (2011)
Ambrosio, L., Mondino, A., Savaré, G.: On the Bakry-Émery condition, the gradient estimates and the local-to-global property of \(\text{ RCD }^{*}(K, N)\) metric measure spaces. J. Geom. Anal. 26, 24–56 (2016)
Ambrosio, L., Mondino, A., Savaré, G.: Nonlinear diffusion equations and curvature conditions in metric measure spaces. arXiv:1509.07273 (2015) (to appear in Mem. Am. Math. Soc.)
Ambrosio, L., Gigli, N., Savaré, G.: Lectures in Mathematics. Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. ETH Zürich, Birkhäuser, Basel (2008)
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, 289–391 (2014)
Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163, 1405–1490 (2014)
Ambrosio, L., Gigli, N., Mondino, A., Rajala, T.: Riemannian Ricci curvature lower bounds in metric spaces with \(\sigma \)-finite measure. Trans. Am. Math. Soc. 367, 4661–4701 (2015)
Bianchini, S., Cavalletti, F.: The Monge problem for distance cost in geodesic spaces. Commun. Math. Phys. 318, 615–673 (2013)
Bianchini, S., Caravenna, L.: On the extremality, uniqueness and optimality of transference plans. Bull. Inst. Math. Acad. Sin. (N.S.) 4, 353–454 (2009)
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)
Calabi, E.: On manifolds with non-negative Ricci-curvature II. Not. Am. Math. Soc. 22, A205 (1975)
Cavalletti, F.: Monge problem in metric measure spaces with Riemannian curvature-dimension condition. Nonlinear Anal. 99, 136–151 (2014)
Cavalletti, F., Huesmann, M.: Existence and uniqueness of optimal transport maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 1367–1377 (2015)
Cavalletti, F., Mondino, A.: Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds. Invent. Math. (2016). doi:10.1007/s00222-016-0700-6
Cavalletti, F., Mondino, A.: Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds. Geom. Topol. 21, 603–645 (2017)
Cavalletti, F., Mondino, A.: Optimal maps in essentially non-branching spaces. Commun. Contemp. Math. (2016). doi:10.1142/S0219199717500079
Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999)
Cheeger, J., Colding, T.H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. 144, 189–237 (1996)
Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46, 406–480 (1997)
Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. II. J. Differ. Geom. 54, 13–35 (2000)
Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom. 54, 37–74 (2000)
Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom. 6, 119–128 (1971/72)
De Philippis, G., Gigli, N.: From volume cone to metric cone in the nonsmooth setting. Geom. Funct. Anal. 26, 1526–1587 (2016)
Erbar, M., Kuwada, K., Sturm, K.T.: On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure space. Invent. Math. 201, 993–1071 (2015)
Fremlin, D.H.: Measure Theory, vol. 4. Torres Fremlin, Colchester (2002)
Garofalo, N., Mondino, A.: Li-Yau and Harnack type inequalities in \(\text{ RCD }^{*}(K, N)\) metric measure spaces. Nonlinear Analysis: Theory, Methods & Applications 95, 721–734 (2014)
Gigli, N.: Optimal maps in non branching spaces with Ricci curvature bounded from below. Geom. Funct. Anal. 22, 990–999 (2012)
Gigli, N.: On the Differential Structure of Metric Measure Spaces and Applications, vol. 236. Memoirs of the American Mathematical Society, Providence (2015)
Gigli, N.: Nonsmooth differential geometry-an approach tailored for spaces with Ricci curvature bounded from below. arXiv:1407.0809 (2014) (to appear in Mem. Am. Math. Soc)
Gigli, N.: The splitting theorem in non-smooth context. arXiv:1302.5555 (2013)
Gigli, N.: An overview of the proof of the splitting theorem in spaces with non-negative Ricci curvature. Anal. Geom. Metr. Spaces 2, 169–213 (2014)
Gigli, N., Han, B.: Sobolev spaces on warped products. arXiv:1512.03177v1 (2015)
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, 2914–2929 (2016)
Gigli, N., Mondino, A.: A PDE approach to nonlinear potential theory in metric measure spaces. J. Math. Pures Appl. 100, 505–534 (2013)
Gigli, N., Mondino, A., Rajala, T.: Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below. J. Reine Angew. Math. 705, 233–244 (2015)
Jiang, Y., Zhang, H.C.: Sharp spectral gaps on metric measure spaces. Calc. Var. PDE 55, 14 (2016)
Ketterer, C.: Cones over metric measure spaces and the maximal diameter theorem. J. Math. Pures Appl. 103, 1228–1275 (2015)
Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169, 903–991 (2009)
Lott, J., Villani, C.: Weak curvature conditions and functional inequalities. J. Funct. Anal. 245, 311–333 (2007)
Mondino, A., Naber, A.: Structure theory of metric-measure spaces with lower Ricci curvature bounds I. arXiv:1405.2222 (2014)
Ohta, Shin-ichi: On the measure contraction property of metric measure spaces. Comment. Math. Helv. 82, 805–828 (2007)
Rajala, T.: Local Poincaré inequalities from stable curvature conditions on metric spaces. Calc. Var. PDE. 44, 477–494 (2012)
Rajala, T.: Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm. J. Funct. Anal. 263, 896–924 (2012)
Rajala, T., Sturm, K.T.: Non-branching geodesics and optimal maps in strong \(\text{ CD }(K,\infty )\)-spaces. Calc. Var. PDE. 50, 831–846 (2014)
Sormani, C.: The almost rigidity of manifolds with lower bounds on Ricci curvature and minimal volume growth. Commun. Anal. Geom. 8, 159–212 (2000)
Sormani, C.: Busemann functions on manifolds with lower bounds on Ricci curvature and minimal volume growth. J. Differ. Geom. 48, 557–585 (1998)
Sturm, K.T.: On the geometry of metric measure spaces I. Acta Math. 196, 65–131 (2006)
Sturm, K.T.: On the geometry of metric measure spaces II. Acta Math. 196, 133–177 (2006)
Villani, C.: Optimal transport, Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)
Yau, S.-T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Math. J. 25, 659–670 (1976)
Zhang, H.C., Zhu, X.P.: Ricci curvature on Alexandrov spaces and rigidity theorems. Commun. Anal. Geom. 18, 503–553 (2010)
Zhang, H.C., Zhu, X.P.: Local Li-Yau’s estimates on \(\text{ RCD }^{*}(K, N)\) metric measure spaces. Calc. Var. PDE 55, 93 (2016)
Acknowledgements
The author would like to thank Prof. X.P. Zhu, B.L. Chen, and H.C. Zhang for their encouragements and helpful discussions. The author is grateful to the anonymous referees for careful reading and giving many valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, XT. Non-compact \(\text {RCD}(0,N)\) Spaces with Linear Volume Growth. J Geom Anal 28, 1005–1051 (2018). https://doi.org/10.1007/s12220-017-9852-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-017-9852-x