Canonical contact forms on spherical CR manifolds

Abstract

We construct the CR invariant canonical contact form can(J) on scalar positive spherical CR manifold (M,J), which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold OHgr(Gamma)/Gamma, where Gamma is a convex cocompact subgroup of AutCRS2n+1=PU(n+1,1) and OHgr(Gamma) is the discontinuity domain of Gamma. This contact form can be used to prove that OHgr(Gamma)/Gamma is scalar positive (respectively, scalar negative, or scalar vanishing) if and only if the critical exponent delta(Gamma)n, or delta(Gamma)=n). This generalizes Nayatanirsquos result for convex cocompact subgroups of SO(n+1,1). We also discuss the connected sum of spherical CR manifolds.