The Yamabe problem on CR manifolds (Q1112988)

The Yamabe problem is to find a metric of constant scalar curvature in a given conformal class of Riemannian metrics. This paper studies the analogous problem for strictly pseudoconvex CR-manifolds. The notion of scalar curvature used here is due to S. Webster. After introducing the relevant variational problem for CR manifolds an invariant \(\lambda\) (N) is defined to be the infimum of the functional under consideration. This corresponds in the Riemannian case to the infimum of the total scalar curvature in a given conformal class with fixed volume. The main result of the paper proves that if \(N^{2n+1}\) is a compact, integrable strictly pseudoconvex CR manifold with \(\lambda (N)<\lambda (S^{2n+1}),\) then the infimum is attained by a solution of constant Webster scalar curvature. The proof is modelled closely on the solution of the Riemannian Yamabe problem in the ``easy'' case due to N. Trudinger and T. Aubin. The analytical estimates are a bit more involved in the CR case and are based on work of G. Folland and E. Stein.