Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem

A CR structure on a real manifold M is a distinguished complex subbundle Z of the complex tangent bundle C TM, satisfying X n Z = 0 and [k, X] c X. For example, the complex structure of Cn+I induces a natural biholomorphically invariant CR structure on any real hypersurface: *' is the space of vectors in the span of a/lz1, . /. ., aaZn+l which are tangent to the hypersurface. An abstract CR manifold M is said to be of hypersurface type if dimR M = 2n + I and dim, X = n; all our CR manifolds will be of this type. If M is oriented, then there is a globally defined real one-form 0 that annihilates *' and . The Levi form, given by Lo (V, W) = -2idO (V A W), is a hermitian form on X. We will assume that the CR structure is strictly pseudoconvex: for some choice of 0, the Levi form Lo is positive definite on Z. In this case 0 defines a contact structure on M and we call 0 a contact form associated with the CR structure. The Levi form plays a role similar to that of the metric in Riemannian geometry. However, the CR structure only determines the Levi form up to a conformal multiple; this multiple is fixed by the choice of a contact form. A CR structure with a given choice of contact form is called a pseudohermitian structure. Thus there is an analogy between pseudohermitian and CR manifolds on the one hand and Riemannian and conformal manifolds on the other. In particular, Webster [W1, W2] and Tanaka [T] have defined a pseudohermitian scalar curvature associated to Lo. The CR Yamabe problem is: given a compact, strictly pseudoconvex CR manifold, find a choice of contact form for which the pseudohermitian scalar curvature is constant. Suppose M is a strictly pseudoconvex CR manifold of dimension 2n + 1. Solutions to the CR Yamabe problem are precisely the critical points of the CR