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Potential Analysis

, Volume 44, Issue 1, pp 27–41 | Cite as

On Measure Contraction Property without Ricci Curvature Lower Bound

  • Paul W. Y. Lee
Article

Abstract

Measure contraction properties M C P (K, N) are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension N, then M C P (K, N) is equivalent to Ricci curvature bounded below by K. On the other hand, it was observed in Rifford (Math. Control Relat. Fields 3(4), 467–487 2013) that there is a family of left invariant metrics on the three dimensional Heisenberg group for which the Ricci curvature is not bounded below. Though this family of metric spaces equipped with the Harr measure satisfy M C P (0,5). In this paper, we give sufficient conditions for a 2n+1 dimensional weakly Sasakian manifold to satisfy M C P (0, 2n + 3). This extends the above mentioned result on the Heisenberg group in Rifford (Math. Control Relat. Fields 3(4), 467–487 2013).

Keywords

Measure contraction property Ricci curvature lower bound Weakly Sasakian manifolds 

Mathematics Subject Classification (2010)

53C21 53C23 

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References

  1. 1.
    Agrachev, A., Lee, P.W.Y.: Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds. Math. Ann. 360(1-2), 209–253 (2014)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84, vol. 1123, pp 177–206. Springer (1985)Google Scholar
  3. 3.
    Blair, D.E.: Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, vol. 509, 146ppGoogle Scholar
  4. 4.
    Coulhon, T., Holopainen, I., Saloff-Coste, L.: Harnack inequality and hyperbolicity for subelliptic p-Laplacians with applications to Picard type theorems. Geom. Funct. Anal. 11(6), 1139–1191 (2001)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Folland, G.B., Stein, E.M.: Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, p xii+285. Princeton University Press, Princeton (1982). University of Tokyo Press, TokyoGoogle Scholar
  6. 6.
    Grigoryan, A.A.: The heat equation on noncompact Riemannian manifolds. (Russian) Mat. Sb. 182(1), 55–87 (1991). translation in Math. USSR-Sb 72(1):47–77 (1992)MathSciNetGoogle Scholar
  7. 7.
    Jerison, D.: The Poincare inequality for vector fields satisfying the Hörmander condition. Duke Math. J 53, 503–523 (1986)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Juillet, N.: Geometric inequalities and generalized Ricci bounds in the Heisenberg group. Int. Math. Res. Not. IMRN 13, 2347–2373 (2009)MathSciNetMATHGoogle Scholar
  9. 9.
    Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. III J. Fac. Sci. Univ. Tokyo Sect. IA Math 34(2), 391–442 (1987)MathSciNetMATHGoogle Scholar
  10. 10.
    Lee, P.W.Y., Li, C., Zelenko, I.: Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds, preprint, arXiv:1304.2658, 25 pp (2013)
  11. 11.
    Lee, P.W.Y.: Generalized Li-Yau estimates and Huisken’s monotonicity formula, arXiv:1211.5559, 25pp, submitted for publication (2013)
  12. 12.
    Lee, P.W.Y.: Differential Harnack inequalities for a family of sub-elliptic diffusion equations on Sasakian manifolds. arXiv: 1302.3315, 27pp, submitted for publication (2013)
  13. 13.
    Levin, J.J.: On the matrix Riccati equation. Proc. Amer. Math. Soc 10, 519–524 (1959)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169(3), 903–991 (2009)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Moser, J.: On Harnacks theorem for elliptic differential equations. Comm. Pure Appl. Math 14, 577–591 (1961)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math 17, 101–134 (1964)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Moser, J.: On pointwise estimate for parabolic differential equations. Comm Pure Appl. Math 24, 727–740 (1971)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Ohta, S.: On the measure contraction property of metric measure spaces. Comment. Math. Helv 82(4), 805–828 (2007)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Ollivier, Y.: Ricci curvature of Markov chains on metric spaces. J. Funct. Anal 256(3), 810–864 (2009)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Rifford, L.: Ricci curvatures in Carnot groups. Math. Control Relat. Fields 3 (4), 467–487 (2013)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Royden, H.L.: Comparison theorems for the matrix Riccati equation. Comm. Pure Appl. Math 41(5), 739–746 (1988)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices 2, 27–38 (1992)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Sturm, K.T.: On the geometry of metric measure spaces. Acta Math 196(1), 65–131 (2006)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Sturm, K.T.: On the geometry of metric measure spaces II. Acta Math 196(1), 133–177 (2006)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Villani, C.: Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, p 973. Springer-Verlag, Berlin (2009)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.The Chinese University of Hong KongShatinHong Kong

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