Tokyo Journal of Mathematics

A Perturbed CR Yamabe Equation on the Heisenberg Group

Takanari SAOTOME

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We will study the CR Yamabe equation for a partially integrable CR structure on the Heisenberg group which is deformed from the standard CR structure. By using the Lyapunov-Schmidt reduction, it is shown that a perturbed standard solution of the CR Yamabe equation is a solution of the deformed CR Yamabe equation, under certain conditions of the deformation. Especially, the deformed CR structure is only partially integrable, in general.

Article information

Tokyo J. Math., Volume 41, Number 2 (2018), 407-432.

First available in Project Euclid: 18 December 2017

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32C20: Normal analytic spaces


SAOTOME, Takanari. A Perturbed CR Yamabe Equation on the Heisenberg Group. Tokyo J. Math. 41 (2018), no. 2, 407--432.

Export citation


  • D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progress in mathematics 203, Birkhäuser, 2002.
  • S. Brendle, Blow-up phenomena for the Yamabe equation, Journal of the AMS 21 (2008), 951–979.
  • S. Brendle and F. C. Marques, Blow-up phenomena for the Yamabe equation II, J. Diff. Geom. 81 (2009), 225–250.
  • S.-S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Ann. of Math. (2) 133 (1974), 219–271.
  • S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in mathematics 246, Birkhäuser, 2006.
  • G. B. Folland and E. M. Stein, Estimates for the $\overline{\partial}_b$-complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522.
  • N. Gamara and R. Yacoub, CR Yamabe conjecture–-the conformally flat case, Pacific J. Math. 201 (2001), 121–175.
  • D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), 167–197.
  • D. Jerison and J. M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1–13.
  • D. Jerison and J. M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom. 29 (1989), 303–343.
  • M. Khuri, F. C. Marques and R. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom. 81 (2009), 143–196.
  • J. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), 37–91.
  • Y. Y. Li and L. Zhang, Compactness of solutions to the Yamabe problem II, Calc. Var. PDE 24 (2005), 185–237.
  • Y. Y. Li and L. Zhang, Compactness of solutions to the Yamabe problem III, J. Funct. Anal. 245 (2007), 438–474.
  • Y. Li and M. Zhu, Yamabe type equations on three dimensional Riemannian manifolds, Comm. Contemp. Math. 1 (1999), 1–50.
  • F. C. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Diff. Geom. 71 (2005), 315–346.
  • F. C. Marques, Blow-up examples for the Yamabe problem, Calc. Var. PDE 36 (2009), 377–397.
  • O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1–52.