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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 $ℌ$ on the complex tangent bundle $ℂTM$ with $ℌ\cap \overline{ℌ}=0$ and [$ℌ,ℌ\right]\subset ℌ$. If M is oriented, there is a globally defined real 1-form $\theta$ that annihilates H and $\overline{H}·$ The Hermitian form ${L}_{0}\left(V,\overline{W}\right)=-2id\theta \left(V\wedge \overline{W}\right)$ 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($\theta \right)$, $\theta$ 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 $\lambda \left(M\right)<\lambda \left({S}^{2n+1}\right)$ 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 $\stackrel{^}{\theta }$ by a CR automorphism of the sphere. As corollary, they prove that the minimum $\lambda \left({S}^{2n+1}\right)$ is $2\pi n\left(n+1\right)$ and is achieved only by constant multiples of $\stackrel{^}{\theta }$ 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 ${K|w+z·\mu +\nu |}^{-n}$ with $K,\nu \in ℂ$, Im $\nu$ $>|\mu |$ 2/4, $\mu \in {ℂ}^{n}·$

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 for solutions of PDE