Skip to main content

Advertisement

SpringerLink
Log in

Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

By using optimal mass transport theory we prove a sharp isoperimetric inequality in \({\textsf {CD}} (0,N)\) metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of n-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of Brendle (Comm Pure Appl Math 2021:13717, 2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.

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

References

  1. Agostiniani, V., Fogagnolo, M., Mazzieri, L.: Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature. Invent. Math. 222(3), 1033–1101 (2020)

    MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)

    MATH  Google Scholar 

  3. Anderson, M.: On the topology of complete manifold of nonnegative Ricci curvature. Topology 3, 41–55 (1990)

    MathSciNet  Google Scholar 

  4. Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11(4), 573–598 (1976)

    MATH  Google Scholar 

  5. Bacher, K., Sturm, K.-T.: Ricci Bounds for Euclidean and Spherical Cones Singular Phenomena and Scaling in Mathematical Models, pp. 3–23. Springer, Cham (2014)

    MATH  Google Scholar 

  6. Balogh, Z.M., Kristály, A.: Equality in Borell-Brascamp-Lieb inequalities on curved spaces. Adv. Math. 339, 453–494 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Brendle, S.: Sobolev inequalities in manifolds with nonnegative curvature. Comm. Pure. Appl. Math. 2021, 13717 (2021)

    Google Scholar 

  8. Brothers, J.E., Ziemer, W.P.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153–179 (1988)

    MathSciNet  MATH  Google Scholar 

  9. Cabré, X., Ros-Oton, X.: Sobolev and isoperimetric inequalities with monomial weights. J. Differ. Equ. 255(11), 4312–4336 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Cabré, X., Ros-Oton, X., Serra, J.: Sharp isoperimetric inequalities via the ABP method. J. Eur. Math. Soc. 18, 2971–2998 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Carron, G.: Euclidean volume growth for complete Riemannian manifolds. Milan J. Math. 88, 455–478 (2020)

    MathSciNet  MATH  Google Scholar 

  12. Carron, G.: Inegalités isopérimétriques de Faber-Krahn et consequences. Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 205–232, Sémin. Congr., 1, Soc. Math. France, Paris (1996)

  13. Cavalletti, F., Mondino, A.: Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds. Invent. Math. 208(3), 803–849 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Chavel, I.: Riemannian Geometry. A Modern Introduction, 2nd edn. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  15. Cheeger, J., Colding, T.H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. (2) 144(1), 189–237 (1996)

    MathSciNet  MATH  Google Scholar 

  16. Cinti, E., Glaudo, F., Pratelli, A., Ros-Oton, X., Serra, J.: Sharp quantitative stability for isoperimetric inequalities with homogeneous weights. Trans. Amer. Math. Soc. 2021, 1 (2021)

    MATH  Google Scholar 

  17. Colding, T.H.: Ricci curvature and volume convergence. Ann. Math. (2) 145(3), 477–501 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Cordero-Erausquin, D., McCann, R.J., Schmuckenschläger, M.: A Riemannian interpolation inequality à la Borell. Brascamp Lieb. Invent. Math. 146, 219–257 (2001)

    MathSciNet  MATH  Google Scholar 

  19. Cordero-Erausquin, D., Nazaret, B., Villani, C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182(2), 307–332 (2004)

    MathSciNet  MATH  Google Scholar 

  20. Coulhon, T., Saloff-Coste, L.: Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana 9(2), 293–314 (1993)

    MathSciNet  MATH  Google Scholar 

  21. Croke, C.: A sharp four-dimensional isoperimetric inequality. Comment. Math. Helv. 59(2), 187–192 (1984)

    MathSciNet  MATH  Google Scholar 

  22. Del Pino, M., Dolbeault, J.: The optimal Euclidean \(L^p\)-Sobolev logarithmic inequality. J. Funct. Anal. 197, 151–161 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Druet, O., Hebey, E., Vaugon, M.: Optimal Nash’s inequalities on Riemannian manifolds: the influence of geometry. Int. Math. Res. Not. IMRN 14, 735–779 (1999)

    MathSciNet  MATH  Google Scholar 

  24. do Carmo, M.P., Xia, C.: Complete manifolds with non-negative Ricci curvature and the Caffarelli-Kohn-Nirenberg inequalities. Compos. Math. 140, 818–826 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Fogagnolo, M., Mazzieri, L.: Minimising hulls, \(p\)-capacity and isoperimetric inequality on complete Riemannian manifolds. (2020). arXiv:2012.09490

  26. Fusco, N.: The classical isoperimetric theorem (2020). https://www.docenti.unina.it/webdocenti-be/allegati/materiale-didattico/377226

  27. Gentil, I.: The general optimal \(L^p\)-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations. J. Funct. Anal. 202(2), 591–599 (2003)

    MathSciNet  MATH  Google Scholar 

  28. Greene, R.E., Wu, H.: \({\cal{C}}^{\infty } \) convex functions and manifolds of positive curvature. Acta Math. 137, 209–245 (1976)

    MathSciNet  MATH  Google Scholar 

  29. Greene, R.E., Wu, H.: \({\cal{C}}^{\infty } \) approximations of convex, subharmonic, and plurisubharmonic functions. Ann. Sci. ENS 12(1), 48–84 (1979)

    MathSciNet  Google Scholar 

  30. Hebey, E.: Nonlinear analysis on manifolds: Sobolev spaces and inequalities. Courant Lecture Notes in Mathematics, 5. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (1999)

  31. Johne, F.: Sobolev inequalities on manifolds with nonnegative Bakry-Émery Ricci curvature, preprint, (2021). arXiv:2103.08496

  32. Ketterer, C.: Cones over metric measure spaces and the maximal diameter theorem. J. Math. Pures Appl. (9) 103(5), 1228–1275 (2015)

    MathSciNet  MATH  Google Scholar 

  33. Klartag, B.: Needle decompositions in Riemannian geometry. Mem. Am. Math. Soc. 249, 1180 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Kleiner, B.: An isoperimetric comparison theorem. Invent. Math. 108(1), 37–47 (1992)

    MathSciNet  MATH  Google Scholar 

  35. Kolesnikov, A.V., Milman, E.: Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds. Am. J. Math. 140(5), 1147–1185 (2018)

    MATH  Google Scholar 

  36. Kristály, A.: Metric measure spaces supporting Gagliardo-Nirenberg inequalities: volume non-collapsing and rigidities. Calc. Var. Part. Differ. Equ. 55(5), 27 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Kristály, A., Ohta, S.: Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications. Math. Ann. 357(2), 711–726 (2013)

    MathSciNet  MATH  Google Scholar 

  38. Ledoux, M.: On manifolds with non-negative Ricci curvature and Sobolev inequalities. Comm. Anal. Geom. 7(2), 347–353 (1999)

    MathSciNet  MATH  Google Scholar 

  39. Lieb, E.H., Loss, M.: Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI (2001)

  40. Liu, G.: 3-manifolds with nonnegative Ricci curvature. Invent. Math. 193(2), 367–375 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Li, P.: Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature. Ann. Math. (2) 124(1), 1–21 (1986)

    MathSciNet  MATH  Google Scholar 

  42. Lott, J., Villani, C.: Ricci curvature for metric measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009)

    MathSciNet  MATH  Google Scholar 

  43. McCann, R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(3), 589–608 (2001)

    MathSciNet  MATH  Google Scholar 

  44. Menguy, X.: Noncollapsing examples with positive Ricci curvature and infinite topological type. Geom. Funct. Anal. 10(3), 600–627 (2000)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  46. Milman, E.: Beyond traditional curvature-dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension. Trans. Am. Math. Soc. 369(5), 3605–3637 (2017)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  48. Mondino, A., Semola, D.: Polya-Szego inequality and Dirichlet \(p\)-spectral gap for non-smooth spaces with Ricci curvature bounded below. J. Math. Pures Appl. 9(137), 238–274 (2020)

    MathSciNet  MATH  Google Scholar 

  49. Muratori, M., Soave, N.: Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds, (2021). arXiv:2103.08240

  50. Ohta, S.: Needle decompositions and isoperimetric inequalities in Finsler geometry. J. Math. Soc. Jpn. 70, 651–693 (2018)

    MathSciNet  MATH  Google Scholar 

  51. Ohta, S.: A semigroup approach to Finsler geometry: Bakry-Ledoux’s isoperimetric inequality. Comm. Anal. Geom. 2016, 1 (2016)

    Google Scholar 

  52. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

  53. Perelman, G.: Manifolds of positive Ricci curvature with almost maximal volume. J. Am. Math. Soc. 7, 299–305 (1994)

    MathSciNet  MATH  Google Scholar 

  54. Petersen, P.: Riemannian Geometry. Graduate Texts in Mathematics, 3rd edn. Springer, Berlin (2016)

    Google Scholar 

  55. Reiris, M.: On Ricci curvature and volume growth in dimension three. J. Differ. Geom. 99(2), 313–357 (2015)

    MathSciNet  MATH  Google Scholar 

  56. Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006)

    MathSciNet  MATH  Google Scholar 

  57. Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006)

    MathSciNet  MATH  Google Scholar 

  58. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)

    MathSciNet  MATH  Google Scholar 

  59. Talenti, G.: Inequalities in rearrangement invariant function spaces. Nonlinear Analysis, Function Spaces and Applications, vol. 5 (Prague, 1994), 177–230. Prometheus, Prague (1994)

  60. Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)

  61. Wang, Y., Zhang, X.: An Alexandroff-Bakelman-Pucci estimate on Riemannian manifolds. Adv. Math. 232, 499–512 (2013)

    MathSciNet  MATH  Google Scholar 

  62. Weil, A.: Sur les surfaces a courbure négative. C. R. Acad. Sci. Paris Sér I Math. 182, 1069–1071 (1926)

    MATH  Google Scholar 

  63. Wulff, G.: Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallflächen. Zeitschr. Kristallogr. 34, 449–530 (1901)

    Google Scholar 

  64. Xia, C.: The Gagliardo-Nirenberg inequalities and manifolds of non-negative Ricci curvature. J. Funct. Anal. 224(1), 230–241 (2005)

    MathSciNet  MATH  Google Scholar 

  65. Zhu, S.-H.: A finiteness theorem for Ricci curvature in dimension three. J. Differ. Geom. 37(3), 711–727 (1993)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank S. Brendle, G. Carron, L. Mazzieri, E. Milman and S. Ohta for stimulating conversations in the early stage of the manuscript. We also would like to thank the Reviewers for their thoughtful comments and efforts towards improving the presentation of our paper.

Author information

Authors and Affiliations

  1. Mathematisches Institute, Universität Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Zoltán M. Balogh

  2. Department of Economics, Babeş-Bolyai University, Str. Teodor Mihali 58-60, 400591, Cluj-Napoca, Romania

    Alexandru Kristály

  3. Institute of Applied Mathematics, Óbuda University, Bécsi út 96/B, 1034, Budapest, Hungary

    Alexandru Kristály

Authors
  1. Zoltán M. Balogh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandru Kristály
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandru Kristály.

Ethics declarations

Conflict of interest

The authors state that there is no conflict of interest.

Additional information

Communicated by Thomas Schick.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Z. M. Balogh was supported by the Swiss National Science Foundation, Grant Nr. 200020_191978. A. Kristály was supported by the UEFISCDI/CNCS grant PN-III-P4-ID-PCE2020-1001.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Balogh, Z.M., Kristály, A. Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature. Math. Ann. 385, 1747–1773 (2023). https://doi.org/10.1007/s00208-022-02380-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-022-02380-1

Mathematics Subject Classification

Search

Navigation