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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 on the complex tangent bundle TM with ¯=0 and [,]. If M is oriented, there is a globally defined real 1-form θ that annihilates H and H ¯· The Hermitian form L 0 (V,W ¯)=-2idθ(VW ¯) is the Levi form. If, for some choice of θ, L θ is positive definite, the CR structure is said to be strictly pseudoconvex. This θ 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(θ), θ 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 λ(M)<λ(S 2n+1 ) for the minimal value λ (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 θ is a contact form associated with the standard CR structure on the sphere which has constant pseudohermitian scalar curvature, then θ is obtained from a constant multiple of the standard form θ ^ by a CR automorphism of the sphere. As corollary, they prove that the minimum λ(S 2n+1 ) is 2πn(n+1) and is achieved only by constant multiples of θ ^ 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 1/2π n 2. Equality is attained only by the functions K|w+z·μ+ν| -n with K,ν, Im ν >|μ| 2/4, μ 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:
32T99Pseudoconvex domains
35B45A priori estimates for solutions of PDE