Skip to main content

Advertisement

SpringerLink
Log in

Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We prove that if \((X,\mathsf {d},\mathfrak {m})\) is a metric measure space with \(\mathfrak {m}(X)=1\) having (in a synthetic sense) Ricci curvature bounded from below by \(K>0\) and dimension bounded above by \(N\in [1,\infty )\), then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by Milman for any \(K\in \mathbb {R}\), \(N\ge 1\) and upper diameter bounds) holds, i.e. the isoperimetric profile function of \((X,\mathsf {d},\mathfrak {m})\) is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume \(v \in (0,1)\) and K is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume \(v \in (0,1)\) and K is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov–Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. During all the paper we will assume \((X,\mathsf {d})\) to be complete, separable and proper

References

  1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs (2000)

  2. Ambrosio, L., Gigli, N., Mondino, A., Rajala, T.: Riemannian Ricci curvature lower bounds in metric measure spaces with \(\sigma \)-finite measure. Trans. Am. Math. Soc. 367(7), 4661–4701 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosio, L., Gigli, N., Savaré, G.: Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43(1), 339–404 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163, 1405–1490 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ambrosio, L., Mondino, A., Savaré, G.: Nonlinear diffusion equations and curvature conditions in metric measure spaces. arXiv:1509.07273

  7. Ambrosio, L., Mondino, A., Savaré, G.: On the Bakry-Émery condition, the gradient estimates and the Local-to-Global property of \(RCD^*(K, N)\) metric measure spaces. J. Geom. Anal. 26, 24–56 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bayle, V.: A differential inequality for the isoperimetric profile. Int. Math. Res. Not. 7, 311–342 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bakry, D., Emery, M.: Diffusions hypercontractives Seminaire de Probabilites XIX. Lecture Notes in Math, vol. 1123, pp. 177–206. Springer-Verlag, New York (1985)

  11. Bakry, D., Ledoux, M.: A logarithmic Sobolev form of the Li-Yau parabolic inequality. Rev. Mat. Iberoam. 22(2), 683–702 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Burago, Y.D., Zalgaller, V.A.: Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 285. Springer, Berlin (1988)

    Google Scholar 

  13. Bianchini, S., Cavalletti, F.: The Monge problem for distance cost in geodesic spaces. Commun. Math. Phys. 318, 615–673 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cavalletti, F.: Monge problem in metric measure spaces with Riemannian curvature-dimension condition. Nonlinear Anal. 99, 136–151 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cavalletti, F.: Decomposition of geodesics in the Wasserstein space and the globalization property. Geom. Funct. Anal. 24, 493–551 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cavalletti, F., Mondino, A.: Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds. arXiv:1505.02061 (to appear in Geometry and Topology)

  17. Cavalletti, F., Mondino, A.: Optimal maps in essentially non-branching spaces. Commun. Contemp. Math. arXiv:1609.00782 (to appear)

  18. Cavalletti, F., Sturm, K.-T.: Local curvature-dimension condition implies measure-contraction property. J. Funct. Anal. 262, 5110–5127 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cheeger, G., Colding, T.H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. 144(1), 189–237 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cheeger, G., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below I. J. Differ. Geom. 45, 406–480 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cheeger, G., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. II. J. Differ. Geom. 54, 13–35 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cheeger, G., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom. 54, 37–74 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cicalese, M., Leonardi, G.P.: A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal. 206(2), 617–643 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Colding, T., Naber, A.: Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. Math. 176 (2012)

  25. Croke, C.B.: An eigenvalue pinching theorem. Invent. Math. 68(2), 253–256 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Eichmair, M., Metzger, J.: Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions. Invent. Math. 194, 591–630 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Erbar, M., Kuwada, Sturm, K.T.: On the equivalence of the entropic curvature-dimension condition and bochner’s inequality on metric measure space. Invent. Math. 201(3), 993–1071 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Evans, L.C.: Partial differential equations, Graduate Studies in Mathematics, vol. 19. AMS (1998)

  29. Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182(1), 167–211 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Fremlin, D.H.: Measure theory, vol. 4. Torres Fremlin (2002)

  31. Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 941–980 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Bérard, P.H., Besson, G., Gallot, S.: Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov [An isoperimetric inequality generalizing the Paul Levy-Gromov inequality]. Invent. Math. 80(2), 295–308 (1985) (French)

  33. Garofalo, N., Mondino, A.: Li-Yau and Harnack type inequalities in \({\sf RCD}^*(K, N)\) metric measure spaces. Nonlinear Anal. Theory Methods Appl. 95, 721–734 (2014)

    Article  MATH  Google Scholar 

  34. Gigli, N.: Optimal maps in non branching spaces with Ricci curvature bounded from below. Geom. Funct. Anal. 22(4), 990–999 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Gigli, N.: The splitting theorem in non-smooth context (2013). arXiv:1302.5555

  36. 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)

  37. Gigli, N., Mondino, A., Rajala, T.: Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below. Journal fur die Reine und Ang Math. 705, 233–244 (2015)

    MathSciNet  MATH  Google Scholar 

  38. 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)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gromov, M.: Metric structures for Riemannian and non Riemannian spaces. Modern Birkhäuser Classics (2007)

  40. Gromov, M., Milman, V.: Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces. Compos. Math. 62(3), 263–282 (1987)

    MathSciNet  MATH  Google Scholar 

  41. Honda, S.: Cheeger constant, \(p\)-Laplacian, and Gromov-Hausdorff convergence (2014). arXiv:1310.0304v3

  42. Kannan, R., Lovász, L., Simonovits, M.: Isoperimetric problems for convex bodies and a localization lemma. Discret. Comput. Geom. 13(3–4), 541–559 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ketterer, C.: Cones over metric measure spaces and the maximal diameter theorem. J. Math. Pures Appl. 103(5), 1228–1275 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Klartag, B.: Needle decomposition in Riemannian geometry. Mem. AMS. arXiv:1408.6322

  45. Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169, 903–991 (2009)

  46. Lovász, L., Simonovits, M.: Random walks in a convex body and an improved volume algorithm. Random Struct. Algorithms 4(4), 359–412 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  47. Maggi, F.: Sets of finite perimeter and geometric variational problems (an introduction to geometric measure theory). Cambridge Studies in Advanced Mathematics (2012)

  48. Milman, E.: Sharp isoperimetric inequalities and model spaces for curvature-dimension-diameter condition. J. Eur. Math. Soc. 17(5), 1041–1078 (2015)

  49. Milman, E., Rotem, L.: Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures. Adv. Math. 262, 867–908 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  50. Mondino, A.,. Naber, A.: Structure theory of metric-measure spaces with lower Ricci curvature bounds I. arXiv:1405.2222

  51. Morgan, F.: Geometric Measure Theory (A Beginner’s Guide), 4th edn. Elsevier/Academic Press, Amsterdam (2009)

    MATH  Google Scholar 

  52. Morgan, F.: In polytopes, small balls about some vertex minimize perimeter. J. Geom. Anal. 17, 97–106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  53. Morgan, F., Ritoré, M.: Isoperimetric regions in cones. Trans. Am. Math. Soc. 354, 2327–2339 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  54. Ohta, S.I.: Finsler interpolation inequalities. Calc. Var. Partial Differ. Equ. 36, 211–249 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  55. Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84(6), 1182–1238 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  56. Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5, 286–292 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  57. Petrunin, A.: Harmonic functions on Alexandrov spaces and their applications. Electron. Res. Announc. AMS 9, 135–141 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  58. Petrunin, A.: Alexandrov meets Lott-Sturm-Villani. Münster J. Math. 4, 53–64 (2011)

    MathSciNet  MATH  Google Scholar 

  59. Rajala, T.: Local Poincaré inequalities from stable curvature conditions on metric spaces. Calc. Var. Partial Differ. Equ. 44, 477–494 (2012)

    Article  MATH  Google Scholar 

  60. Rajala, T., Sturm, K.T.: Non-branching geodesics and optimal maps in strong \({\sf CD}(K,\infty )\)-spaces. Calc. Var. Partial Differ. Equ. 50, 831–846 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  61. Ritoré, M.: Geometric flows, isoperimetric inequalities and hyperbolic geometry, Mean curvature flow and isoperimetric inequalities. Adv. Courses Math. CRM Barcelona, pp. 45-113. Birkhäuser, Basel (2010)

  62. Ros, A.: The isoperimetric problem, Lecture series at the Clay Mathematics Institute, Summer School on the Global Theory of Minimal Surfaces, MSRI, Berkeley, California (2001)

  63. Savaré, G.: Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD\((K,\infty )\) metric measure spaces. Discret. Cont. Dyn. Sist. A 34, 1641–1661 (2014)

    Article  MATH  Google Scholar 

  64. Sturm, K.T.: On the geometry of metric measure spaces. I. Acta Math. 196, 65–131 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  65. Sturm, K.T.: On the geometry of metric measure spaces. II. Acta Math. 196, 133–177 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  66. Villani, C.: Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer-Verlag, Berlin (2009)

  67. Zhang, H.C., Zhu, X.P.: Ricci curvature on Alexandrov spaces and rigidity theorems. Commun. Anal. Geom. 18(3), 503–553 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank Emanuel Milman for having drawn their attention to the recent paper by Klartag [44] and the reviewers, whose detailed comments led to an improvement of the manuscript. They also wish to thank the Hausdorff center of Mathematics of Bonn, where most of the work has been developed, for the excellent working conditions and the stimulating atmosphere during the trimester program “Optimal Transport” in Spring 2015. The second author gratefully acknowledges the support of the ETH-fellowship.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Pavia, Pavia, Italy

    Fabio Cavalletti

  2. Institut für Mathematik, ETH-Zürich and Universität Zürich, Zurich, Switzerland

    Andrea Mondino

Authors
  1. Fabio Cavalletti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrea Mondino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabio Cavalletti.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cavalletti, F., Mondino, A. Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds. Invent. math. 208, 803–849 (2017). https://doi.org/10.1007/s00222-016-0700-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-016-0700-6

Search

Navigation