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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 ${\frak H}$ on the complex tangent bundle ${\bbfC}TM$ with ${\frak H}\cap\bar {\frak H}=0$ and [${\frak H},{\frak H}]\subset {\frak H}$. If M is oriented, there is a globally defined real 1-form $\theta$ that annihilates H and $\bar H.$ The Hermitian form $L\sb 0(V,\bar W)=- 2id\theta (V\wedge \bar W)$ is the Levi form. If, for some choice of $\theta$, $L\sb{\theta}$ 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 $\lambda (M)<\lambda (S\sp{2n+1})$ 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 $\lambda (S\sp{2n+1})$ 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 {\it G. B. Folland} and {\it 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 $K\vert w+z\cdot \mu +\nu \vert\sp{-n}$ with $K,\nu\in {\bbfC}$, Im $\nu$ $>\vert \mu \vert$ 2/4, $\mu\in {\bbfC}\sp n.$ In the proof of the theorem, they use the idea of {\it 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

32T99Pseudoconvex domains
35B45A priori estimates for solutions of PDE
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