*(English)*Zbl 0634.32016

A CR structure on a real manifold M is a distinguished complex subbundle $\u210c$ on the complex tangent bundle $\u2102TM$ with $\u210c\cap \overline{\u210c}=0$ and [$\u210c,\u210c]\subset \u210c$. If M is oriented, there is a globally defined real 1-form $\theta $ that annihilates H and $\overline{H}\xb7$ The Hermitian form ${L}_{0}(V,\overline{W})=-2id\theta (V\wedge \overline{W})$ 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 $\lambda \left(M\right)<\lambda \left({S}^{2n+1}\right)$ 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 $\widehat{\theta}$ by a CR automorphism of the sphere. As corollary, they prove that the minimum $\lambda \left({S}^{2n+1}\right)$ is $2\pi n(n+1)$ and is achieved only by constant multiples of $\widehat{\theta}$ 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 ${K|w+z\xb7\mu +\nu |}^{-n}$ with $K,\nu \in \u2102$, Im $\nu $ $>\left|\mu \right|$ 2/4, $\mu \in {\u2102}^{n}\xb7$

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.