## Tokyo Journal of Mathematics

### A Perturbed CR Yamabe Equation on the Heisenberg Group

Takanari SAOTOME

#### Abstract

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

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

Dates
First available in Project Euclid: 18 December 2017

https://projecteuclid.org/euclid.tjm/1513566021

Mathematical Reviews number (MathSciNet)
MR3908802

Zentralblatt MATH identifier
07053484

Subjects
Primary: 32C20: Normal analytic spaces

#### Citation

SAOTOME, Takanari. A Perturbed CR Yamabe Equation on the Heisenberg Group. Tokyo J. Math. 41 (2018), no. 2, 407--432. https://projecteuclid.org/euclid.tjm/1513566021

#### References

• 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.