Mathematische Annalen

, Volume 373, Issue 1–2, pp 743–830 | Cite as

Convergence of the CR Yamabe flow

  • Pak Tung HoEmail author
  • Weimin Sheng
  • Kunbo Wang


We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold M of real dimension \(2n+1\). We prove convergence of the CR Yamabe flow when \(n=1\) or M is spherical.

Mathematics Subject Classification

Primary 32V20 53C44 Secondary 53C21 35R03 



The first author would like to thank Prof. Paul Yang, who encouraged him to work on the general convergence of the CR Yamabe flow, Prof. Simon Brendle who answered many of his questions, and Prof. Sai-Kee Yeung for helpful discussions. The second and the third authors would also like to thank Prof. Paul Yang, who brought them into the field of CR geometry since his short course at Zhejiang University in June of 2014. Part of the work was done when the first author visited Princeton University and Zhejiang University, and he is grateful for their kind hospitality.


  1. 1.
    Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. (9) 55, 269–296 (1976)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bahri, A., Coron, J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Commun. Pure Appl. Math. 41, 253–294 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bramanti, M., Brandolini, L.: Schauder estimates for parabolic nondivergence operators of Hörmander type. J. Differ. Equ. 234, 177–245 (2007)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bramanti, M., Brandolini, L.: \(L^p\) estimate for nonvariational hypoelliptic operators with VMO coefficients. Trans. Am. Math. Soc. 352, 781–822 (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Brendle, S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differ. Geom. 69, 217–278 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brendle, S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541–576 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chang, S.C., Cheng, J.H.: The Harnack estimate for the Yamabe flow on CR manifolds of dimension 3. Ann. Glob. Anal. Geom. 21, 111–121 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chang, S.C., Chiu, H.L., Wu, C.T.: The Li-Yau-Hamilton inequality for Yamabe flow on a closed CR 3-manifold. Trans. Am. Math. Soc. 362, 1681–1698 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chanillo, S., Chiu, H.L., Yang, P.: Embeddability for 3-dimensional Cauchy–Riemann manifolds and CR Yamabe invariants. Duke Math. J. 161, 2909–2921 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cheng, J.H., Chiu, H.L., Yang, P.: Uniformization of spherical CR manifolds. Adv. Math. 255, 182–216 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cheng, J.H., Malchiodi, A., Yang, P.: A positive mass theorem in three dimensional Cauchy–Riemann geometry. Adv. Math. 308, 276–347 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chow, B.: The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Commun. Pure Appl. Math. 45, 1003–1014 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Citti, G.: Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian. Ann. Mat. Pura Appl. (4) 169, 375–392 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Citti, G., Uguzzoni, F.: Critical semilinear equations on the Heisenberg group: the effect of the topology of the domain. Nonlinear Anal. 46, 399–417 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics. 246. Birkhäuser Boston, Boston (2006)zbMATHGoogle Scholar
  16. 16.
    Folland, G.: The tangential Cauchy–Riemann complex on spheres. Trans. Am. Math. Soc. 171, 83–133 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Folland, G., Stein, E.: Estimates for the \({\bar{\partial }} _{b}\) complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gamara, N.: The CR Yamabe conjecture—the case \(n=1\). J. Eur. Math. Soc. 3, 105–137 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gamara, N., Yacoub, R.: CR Yamabe conjecture—the conformally flat case. Pac. J. Math. 201, 121–175 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Habermann, L.: Conformal metrics of constant mass. Calc. Var. Partial Differ. Equ. 12, 259–279 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Habermann, L., Jost, J.: Green functions and conformal geometry. J. Differ. Geom. 53, 405–433 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. Am. Math. Soc. (Providence, RI) 71, 237–262 (1988)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ho, P.T.: Result related to prescribing pseudo-Hermitian scalar curvature. Int. J. Math. 24, 29 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ho, P.T.: The long time existence and convergence of the CR Yamabe flow. Commun. Contemp. Math. 14, 50 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ho, P.T.: The Webster scalar curvature flow on CR sphere. Part I. Adv. Math. 268, 758–835 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jerison, D., Lee, J.M.: Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. J. Am. Math. Soc. 1, 1–13 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Jerison, D., Lee, J.M.: Intrinsic CR normal coordinates and the CR Yamabe problem. J. Differ. Geom. 29, 303–343 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jerison, D., Lee, J.M.: The Yamabe problem on CR manifolds. J. Differ. Geom. 25, 167–197 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lee, J.M.: The Fefferman metric and pseudo-Hermitian invariants. Trans. Am. Math. Soc. 296, 411–429 (1986)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. (N.S.) 17, 37–91 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Schwetlick, H., Struwe, M.: Convergence of the Yamabe flow for “large” energies. J. Reine Angew. Math. 562, 59–100 (2003)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Simon, L.: Asymptotics for a class of non-linear evolution equations with applications to geometric problems. Ann. Math. (2) 118, 525–571 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511–517 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Tanaka, N.: A differential geometric study on strongly pseudo-convex manifolds. Kinokuniya, Tokyo (1975)zbMATHGoogle Scholar
  36. 36.
    Trudinger, N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3) 22, 265–274 (1968)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Wang, W.: Canonical contact forms on spherical CR manifolds. J. Eur. Math. Soc. 5, 245–273 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Webster, S.M.: Pseudohermitian structures on a real hypersurface. J. Differ. Geom. 13, 25–41 (1978)CrossRefzbMATHGoogle Scholar
  39. 39.
    Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Ye, R.: Global existence and convergence of Yamabe flow. J. Differ. Geom. 39, 35–50 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zhang, Y.: The contact Yamabe flow. Ph.D. thesis, University of Hanover (2006)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsSogang UniversitySeoulKorea
  2. 2.School of Mathematical SciencesZhejiang UniversityHangzhouChina

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