Skip to main content
SpringerLink
Log in

On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Émery (via energy and \(\Gamma _2\)-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the \(L^2\)-Wasserstein distance.

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. Ambrosio, L., Gigli, N.: User’s Guide to Optimal Transport Theory. CIME Lecture Notes in Mathematics (2012)

  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. (preprint)

  3. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008)

    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, 289–391 (2013)

    Article  Google Scholar 

  5. Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163(7), 1405–1490 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

  7. Ambrosio, L., Mondino, A., Savaré, G.: Nonlinear diffusion equations and curvature conditions in metric measure spaces (in preparation)

  8. 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. (2014, to appear)

  9. Bacher, K., Sturm, K.-T.: Localization and tensorization properties of the curvature-dimension condition for metric measure spaces. J. Funct. Anal. 259(1), 28–56 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bacher, K., Sturm, K.-T.: Ricci bounds for euclidean and spherical cones. In: Griebel, M. (ed.) Singular Phenomena and Scaling in Mathematical Models, pp. 3–23. Springer International Publishing, Switzerland (2014)

  11. Bakry, D.: L’hypercontractivité et son utilisation en théorie des semigroupes. In: Bernard, P. (ed.) Lectures on Probability Theory (Saint-Flour, 1992). Lecture Notes in Mathematics, vol. 1581, pp. 1–114. Springer, Berlin (1994)

  12. Bakry, D., Émery, M.: Diffusions hypercontractives. In: Azéma, J., Yor, M. (eds.) Séminaire de probabilités, XIX, 1983/84. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985)

  13. 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 

  14. Bolley, F., Gentil, I., Guillin, A.: Dimensional contraction via Markov transportation distance . J. Lond. Math. Soc. 90(1), 309–332 (2014)

  15. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)

  16. Cavalletti, F., Sturm, K.-T.: Local curvature-dimension condition implies measure-contraction property. J. Funct. Anal. 262(12), 5110–5127 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Daneri, S., Savaré, G.: Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math. Anal. 40(3), 1104–1122 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1990)

  19. Dembo, A., Cover, T.M., Thomas, J.: Information-theoretic inequalities. IEEE Trans. Inf. Theory 37(6), 1501–1518 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Garofalo, N., Mondino, A.: Li–Yau and Harnack type inequalities in metric measure spaces. Nonlinear Anal. 95, 721–734 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gigli, N.: On the differential structure of metric measure spaces and applications (preprint). arXiv:1205.6622

  22. Gigli, N.: On the heat flow on metric measure spaces: existence, uniqueness and stability. Calc. Var. Partial Differ. Equ. 39(1–2), 101–120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gigli, N., Kuwada, K., Ohta, S.-I.: Heat flow on Alexandrov spaces. Commun. Pure Appl. Math. 66(3), 307–331 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), x+101 (2000)

  25. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ketterer, C.: Ricci curvature bounds for warped products. J. Funct. Anal. 265(2), 266–299 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Koskela, P., Zhou, Y.: Geometry and analysis of Dirichlet forms. Adv. Math. 231(5), 2755–2801 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kuwada, K.: Space-time Wasserstein control and Bakry–Ledoux type gradient estimates. Calc. Var. Partial Differ. Equ. (2014, to appear)

  29. Kuwada, K.: Duality on gradient estimates and Wasserstein controls. J. Funct. Anal. 258(11), 3758–3774 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)

    Article  MathSciNet  Google Scholar 

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

  32. Naber, A.: Characterizations of bounded Ricci curvature on smooth and nonsmooth spaces (preprint). arXiv:1306.6512

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

    Article  MATH  Google Scholar 

  34. Ohta, S.-I., Sturm, K.-T.: Bochner–Weitzenböck formula and Li–Yau esitmate on Finsler manifolds . Adv. Math. 252, 429–448 (2014)

  35. Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2), 361–400 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Otto, F., Westdickenberg, M.: Eulerian calculus for the contraction in the Wasserstein distance. SIAM J. Math. Anal. 37(4), 1227–1255 (2005)

  37. Petrunin, A.: Alexandrov meets Lott–Villani–Sturm. Münst. J. Math. 4, 53–64 (2011)

    MathSciNet  MATH  Google Scholar 

  38. Profeta, A.: The sharp Sobolev inequality on metric measure spaces with lower Ricci curvature bounds (preprint)

  39. Rajala, T.: Local Poincaré inequalities from stable curvature conditions on metric spaces. Calc. Var. Partial Differ. Equ. 44(3–4), 477–494 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rajala, T., Sturm, K-Th: Non-branching geodesics and optimal maps in strong CD\((K,{\infty })\)-spaces. Calc. Var. Partial Differ. Equ. 50(3–4), 831–846 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Simon, B.: Convexity. An Analytic Viewpoint. Cambridge Tracts in Mathematics, vol. 187. Cambridge University Press, Cambridge (2011)

  42. Sturm, K.-T.: On the geometry of metric measure spaces. I and II. Acta Math. 196(1), 65–177 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Villani, C.: A short proof of the “concavity of entropy power”. IEEE Trans. Inf. Theory 46(4), 1695–1696 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

  45. Wang, F.-Y.: Equivalent semigroup properties for the curvature-dimension condition. Bull. Sci. Math. 135(6–7), 803–815 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  47. Zhang, H.-C., Zhu, X.-P.: Yau’s gradient estimates on Alexandrov spaces. J. Differ. Geom. 91(3), 445–522 (2012)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Matthias Erbar & Karl-Theodor Sturm

  2. Graduate School of Science, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo, 152-8551, Japan

    Kazumasa Kuwada

Authors
  1. Matthias Erbar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kazumasa Kuwada
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Karl-Theodor Sturm
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karl-Theodor Sturm.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Erbar, M., Kuwada, K. & Sturm, KT. On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Invent. math. 201, 993–1071 (2015). https://doi.org/10.1007/s00222-014-0563-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-014-0563-7

Search

Navigation