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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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