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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
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##### 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, Tôhoku 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
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