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 subbun-dIe Jf' of the complex tangent bundle CC TM, satisfying Jf' n Jf' = 0 and Jf', Jf' c Jf'.For example, the complex structure of CC n + 1 induces a natural biholomorphically invariant CR structure on any real hypersurface: Jf' is the space of vectors in the span of 8j8z l , ... ,8j8z n + 1 which are tangent to the hypersurface.An abstract CR manifold M is said to be of hypersurface type if dimji M = 2n + 1 and dime Jf' = 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 Jf' and Jf'.The Levi form, given by Lis a hermitian form on Jf'.We will assume that the CR structure is strictly pseudoconvex: for some choice of 0, the Levi form L() is positive definite on Jf'.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 WI, W2 and Tanaka T have defined a pseudohermitian scalar curvature associated to L 9 • 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