Monotonicity and rigidity of the \({\mathcal {W}}\)-entropy on \({\mathsf {RCD}} (0,N)\) spaces


By means of a space-time Wasserstein control, we show the monotonicity of the \({\mathcal {W}}\)-entropy functional in time along heat flows on possibly singular metric measure spaces with non-negative Ricci curvature and a finite upper bound of dimension in an appropriate sense. The associated rigidity result on the rate of dissipation of the \({\mathcal {W}}\)-entropy is also proved. These extend known results even on weighted Riemannian manifolds in some respects. In addition, we reveal that some singular spaces will exhibit the rigidity models while only the Euclidean space does in the class of smooth weighted Riemannian manifolds.

This is a preview of subscription content, access via your institution.


  1. 1.

    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, 4661–4701 (2015)

    MathSciNet  Google Scholar 

  2. 2.

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

    Google Scholar 

  3. 3.

    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)

    MathSciNet  Google Scholar 

  4. 4.

    Ambrosio, L., Gigli, N., Savaré, G.: Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoam. 29, 969–996 (2013)

    MathSciNet  Google Scholar 

  5. 5.

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

    MathSciNet  Google Scholar 

  6. 6.

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

    MathSciNet  Google Scholar 

  7. 7.

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

  8. 8.

    Ambrosio, L., Mondino, A., Savaré, G.: On the Bakry–Émery condition, the gradient estimates and the local-to-global property of metric measure spaces. J. Geom. Anal. 26, 24–56 (2016)

    MathSciNet  Google Scholar 

  9. 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)

    MathSciNet  Google Scholar 

  10. 10.

    Bakry, D., Émery, M.: Diffusions Hypercontractives. Séminaire de probabilités, XIX, 1983/1984. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985)

    Google Scholar 

  11. 11.

    Bakry, D., Gentil, I., Ledoux, M.: On Harnack inequalities and optimal transport. Ann. Sci. Norm. Super. Pisa Cl. Sci. 14, 705–727 (2015)

    MathSciNet  Google Scholar 

  12. 12.

    Balogh, Z.M., Engoulatov, A., Hunziker, L., Maasalo, O.E.: Functional inequalities and Hamilton–Jacobi equations in geodesic spaces. Potential Anal. 36(2), 317–337 (2012)

    MathSciNet  Google Scholar 

  13. 13.

    Baudoin, F., Garofalo, N.: Perelman’s entropy and doubling property on Riemannian manifolds. J. Geom. Anal. 21, 1119–1131 (2011)

    MathSciNet  Google Scholar 

  14. 14.

    Cavalletti, F., Milman, E.: The globalization theorem for the curvature dimension condition. Preprint. arXiv:1612.07623

  15. 15.

    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(3), 993–1071 (2015)

    MathSciNet  Google Scholar 

  16. 16.

    Fang, F., Li, X.-D., Zhang, Z.-L.: Two generalizations of Cheeger–Gromoll spliting theorem via Bakry–Emery Ricci curvature. Annales de l’Institut de Fourier. Tome 59(2), 563–573 (2009)

    Google Scholar 

  17. 17.

    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics, vol. 19. Walter de Gruyter & Co., Berlin (1994)

    Google Scholar 

  18. 18.

    Gigli, N.: Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below. Mem. Am. Math. Soc. 251, 1196 (2018)

    MathSciNet  Google Scholar 

  19. 19.

    Gigli, N.: The splitting theorem in non-smooth context. Preprint. arXiv:1302.5555

  20. 20.

    Gigli, N.: On the differential structure of metric measure spaces and applications. Mem. Am. Math. Soc. 236, vi+91 (2015)

    MathSciNet  Google Scholar 

  21. 21.

    Gigli, N., De Philippis, G.: From volume cone to metric cone in the nonsmooth setting. Geom. Funct. Anal. 26(6), 1526–1587 (2016)

    MathSciNet  Google Scholar 

  22. 22.

    Gigli, N., Mondino, A., Rajala, T.: Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below. J. Reine Angew. Math. 705, 233–244 (2015)

    MathSciNet  Google Scholar 

  23. 23.

    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, 1071–1129 (2015)

    MathSciNet  Google Scholar 

  24. 24.

    Jiang, R.: The Li–Yau inequality and heat kernels on metric measure spaces. J. Math. Pures Appl. (9) 104(1), 29–57 (2015)

    MathSciNet  Google Scholar 

  25. 25.

    Jiang, R., Li, H.-Q., Zhang, H.-C.: Heat kernel bounds on metric measure spaces and some applications. Potential Anal. 44(3), 601–627 (2016)

    MathSciNet  Google Scholar 

  26. 26.

    Jiang, R., Zhang, H.-C.: Hamilton’s gradient estimates and a monotonicity formula for heat flows on metric measure spaces. Nonlinear Anal. 131, 32–47 (2016)

    MathSciNet  Google Scholar 

  27. 27.

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

    MathSciNet  Google Scholar 

  28. 28.

    Kopfer, E., Sturm, K.-T.: Heat flows on time-dependent metric measure spaces and super-Ricci flows. Commun. Pure Appl. Math. 71(2), 2500–2608 (2018)

    MathSciNet  Google Scholar 

  29. 29.

    Kuwada, K.: Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates. Calc. Var. Part. Differ. Equ. 54, 127–161 (2015)

    MathSciNet  Google Scholar 

  30. 30.

    Li, H.-Q.: Sharp heat kernel bounds and entropy in metric measure spaces. Sci. China Math. 61(3), 487–510 (2018)

    MathSciNet  Google Scholar 

  31. 31.

    Li, S., Li, X.-D.: Harnack inequalities and W-entropy formula for Witten Laplacian on Riemannian manifolds with \(\rm K\)-super Perelman Ricci flow. Preprint. arXiv:1412.7034

  32. 32.

    Li, S., Li, X.-D.: W-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds. Preprint. arXiv:1604.02596

  33. 33.

    Li, S., Li, X.-D.: The W-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials. Pacific J. Math. 278(1), 173–199 (2015)

    MathSciNet  Google Scholar 

  34. 34.

    Li, S., Li, X.-D.: Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds. J. Funct. Anal. 274, 3263–3290 (2018)

    MathSciNet  Google Scholar 

  35. 35.

    Li, S., Li, X.-D.: \(W\)-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein spaces over Riemannian manifolds. Sci. China Math. 61, 1385–1406 (2018).

    MathSciNet  Article  Google Scholar 

  36. 36.

    Li, S., Li, X.-D.: On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows. Asian J. Math. 22(3), 577–598 (2018)

    MathSciNet  Google Scholar 

  37. 37.

    Li, X.-D.: Perelman’s W-entropy for the Fokker–Planck equation over complete Riemannian manifolds. Bull. Sci. Math. 135(6–7), 871–882 (2011)

    MathSciNet  Google Scholar 

  38. 38.

    Li, X.-D.: Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature. Math. Ann. 353, 403–437 (2012)

    MathSciNet  Google Scholar 

  39. 39.

    Li, X.-D.: From the Boltzmann H-theorem to Perelman’s W-entropy formula for the Ricci flow. In: Emerging Topics on Differential Equations and Their Applications (Hackensack, NJ), Nankai Ser. Pure Appl. Math. Theoret. Phys., vol. 10, pp. 68–84. World Scientific Publication (2013)

  40. 40.

    Li, X.-D.: Hamilton’s Harnack inequality and the W-entropy formula on complete Riemannian manifolds. Stoch. Process. Appl. 126(4), 1264–1283 (2016)

    MathSciNet  Google Scholar 

  41. 41.

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

    MathSciNet  Google Scholar 

  42. 42.

    Ni, L.: Addenda to "The entropy formula for linear heat equation". J. Geom. Anal. 14, 369–374 (2004)

    MathSciNet  Google Scholar 

  43. 43.

    Ni, L.: The entropy formula for linear heat equation. J. Geom. Anal. 14, 87–100 (2004)

    MathSciNet  Google Scholar 

  44. 44.

    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint. arXiv:0710.3174

  45. 45.

    Rothaus, O.S.: Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators. J. Funct. Anal. 42(1), 110–120 (1981)

    MathSciNet  Google Scholar 

  46. 46.

    Savaré, G.: Self-improvement of the Bakry–Émery condition and Wasserstein contraction of the heat flow in RCD(\(K,\infty \)) metric measure spaces. Discrete Contin. Dyn. Syst. 34(4), 1641–1661 (2014)

    MathSciNet  Google Scholar 

  47. 47.

    Sturm, K.-T.: Super-Ricci flows for metric measure spaces. J. Funct. Anal. 275(12), 3504–3569 (2018)

    MathSciNet  Google Scholar 

  48. 48.

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

    MathSciNet  Google Scholar 

  49. 49.

    Topping, P.: \(\cal{L}\)-optimal transportation for Ricci flow. J. Reine Angew. Math. 636, 93–122 (2009)

    MathSciNet  Google Scholar 

  50. 50.

    Villani, C.: Topics in Optimal Transportations. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)

    Google Scholar 

Download references


This work was started when the authors attended the Workshop on Geometry and Probability held at the Luxembourg University in October 2013. The authors would like to thank Anton Thalmaier for his invitations which lead this work possible. The first author warmly thanks to Shouhei Honda for his suggestion to formulate Theorem 5.8 and to Nicola Gigli for an improvement of Theorem 5.8. He also wish to tell his gratitude to Karl-Theodor Sturm for fruitful discussions. Especially, Theorem 5.6 comes from a discussion with him. The first author was supported by the European Union through the ERC–AdG “Ricci Bounds” for Prof. K. T. Sturm. The second author would like to thank Songzi Li for fruitful collaboration on the study of the W-entropy for heat flows, geodesic and Langevin flows on the Wasserstein space over manifolds. He also would like to express his gratitude to Kazuhiro Kuwae and Yu–Zhao Wang for valuable discussions on various topics related to this work.

Author information



Corresponding author

Correspondence to Xiang-Dong Li.

Additional information

Publisher's Note

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

K. Kuwada: Supported in part by JSPS Grant-in-Aid for Young Scientist (A) (KAKENHI) 26707004.

X.-D. Li: Research supported by NSFC No. 11771430, Key Laboratory RCSDS, CAS, No. 2008DP173182, and Hua Luo-Keng Research Grant of AMSS, CAS (2015-2017).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kuwada, K., Li, XD. Monotonicity and rigidity of the \({\mathcal {W}}\)-entropy on \({\mathsf {RCD}} (0,N)\) spaces. manuscripta math. 164, 119–149 (2021).

Download citation

Mathematics Subject Classification (2010)

  • Primary 53C23
  • 53C44
  • 53J35
  • Secondary 51f99
  • 58J65
  • 60J60