# zbMATH — the first resource for mathematics

Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. (English) Zbl 0634.32016
A CR structure on a real manifold M is a distinguished complex subbundle $${\mathfrak H}$$ on the complex tangent bundle $${\mathbb{C}}TM$$ with $${\mathfrak H}\cap\bar {\mathfrak H}=0$$ and [$${\mathfrak H},{\mathfrak H}]\subset {\mathfrak H}$$. If M is oriented, there is a globally defined real 1-form $$\theta$$ that annihilates H and $$\bar H.$$ The Hermitian form $$L_ 0(V,\bar W)=- 2id\theta (V\wedge \bar W)$$ is the Levi form. If, for some choice of $$\theta$$, $$L_{\theta}$$ is positive definite, the CR structure is said to be strictly pseudoconvex. This $$\theta$$ is called a contact form. Yamabe posed the following problem: given a compact strictly pseudoconvex CR manifold, find a choice of contact form for which the pseudohermitian scalar curvature is constant. Solutions to the CR Yamabe problem are precisely the critical points of the CR Yamabe functional Y($$\theta)$$, $$\theta$$ being any contact form. In the previous paper [J. Diff. Geom. 25, 167-197 (1987)], the authors proved that the problem has a solution provided $$\lambda (M)<\lambda (S^{2n+1})$$ for the minimal value $$\lambda$$ (M) of the functional. The unicity of the solution on the sphere was also conjectured. Their purpose is to confirm this conjecture. They prove the following theorem: If $$\theta$$ is a contact form associated with the standard CR structure on the sphere which has constant pseudohermitian scalar curvature, then $$\theta$$ is obtained from a constant multiple of the standard form $${\hat \theta}$$ by a CR automorphism of the sphere. As corollary, they prove that the minimum $$\lambda (S^{2n+1})$$ is $$2\pi n(n+1)$$ and is achieved only by constant multiples of $${\hat \theta}$$ and its images under CR automorphisms.
This result is equivalently formulated on the Heisenberg group H n. By G. B. Folland and E. Stein [Commun. Pure Appl. Math. 27, 429- 522 (1974; Zbl 0293.35012)], there is a positive constant C such that the Sobolev-type inequality holds for all functions. They prove that the best constant C in the inequality is 1/2$$\pi$$ n 2. Equality is attained only by the functions $$K| w+z\cdot \mu +\nu |^{-n}$$ with $$K,\nu\in {\mathbb{C}}$$, Im $$\nu$$ $$>| \mu |$$ 2/4, $$\mu\in {\mathbb{C}}^ n.$$
In the proof of the theorem, they use the idea of M. Obata’s proof [J. Diff. Geom. 6, 247-258 (1971; Zbl 0236.53042)] of the analogous result in the Riemannian geometry.
Reviewer: J.Kajiwara

##### MSC:
 32T99 Pseudoconvex domains 35B45 A priori estimates in context of PDEs
Full Text:
##### References:
 [1] Eric Bedford, (\partial \partial )_\? and the real parts of CR functions, Indiana Univ. Math. J. 29 (1980), no. 3, 333 – 340. · Zbl 0441.32008 · doi:10.1512/iumj.1980.29.29024 · doi.org [2] Eric Bedford and Paul Federbush, Pluriharmonic boundary values, TĂ´hoku Math. J. (2) 26 (1974), 505 – 511. · Zbl 0298.31012 · doi:10.2748/tmj/1178241074 · doi.org [3] G. B. Folland and E. M. Stein, Estimates for the \partial _\? complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429 – 522. · Zbl 0293.35012 · doi:10.1002/cpa.3160270403 · doi.org [4] David Jerison and John M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Microlocal analysis (Boulder, Colo., 1983) Contemp. Math., vol. 27, Amer. Math. Soc., Providence, RI, 1984, pp. 57 – 63. · doi:10.1090/conm/027/741039 · doi.org [5] David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167 – 197. · Zbl 0661.32026 [6] -, Intrinsic CR normal coordinates and the CR Yamabe problem, preprint, MSRI 321-388, 1987. [7] John M. Lee, The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc. 296 (1986), no. 1, 411 – 429. · Zbl 0595.32026 [8] John M. Lee, Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988), no. 1, 157 – 178. · Zbl 0638.32019 · doi:10.2307/2374543 · doi.org [9] John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37 – 91. · Zbl 0633.53062 [10] Morio Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247 – 258. · Zbl 0236.53042 [11] Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479 – 495. · Zbl 0576.53028 [12] Noboru Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya Book-Store Co., Ltd., Tokyo, 1975. Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9. · Zbl 0331.53025 [13] S. Webster, Real hypersurfaces in complex space, Thesis, University of California, Berkeley, 1975. [14] S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), no. 1, 25 – 41. · Zbl 0379.53016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.