On spherical CR manifolds with positive Webster scalar curvature.

Abstract

This dissertation is motivated by the attempt to complete the CR Yamabe problem. Indeed, the results of this dissertation could be applied to completely resolve the CR Yamabe problem on spherical CR manifolds. The method of this work is subelliptic potential theoretic. Let M be a CR spherical manifold with positive Webster scalar curvature, we classify its universal covering M by showing that M is either CR equivalent to an open domain in complex unit sphere or sphere itself. An argument called "comparing Green's functions" inspired by the work of R. Schoen and S. T. Yau on locally conformally flat geometry is used. More specifically, we look at CR invariant laplacina on M, its Dirichlet problem with solution continuous up to boundary is solved. The results are then applied to construct Green's function on M, and we further demonstrate its growth properties. To conclude the uniformization of M, we prove a singular, subelliptic version "Liouville Property". The classification of M is then applied to study asymptotically Heisenberg structure of M. The local expansion of contact form is explicitly obtained in terms of a geometric data called "CR mass"; A positive mass theorem in this CR spherical setting is proved. We finally apply the asymptotically Heisenberg structure and the CR positive mass theorem to study CR Yamabe problem on spherical CR manifolds. We estimate the CR Yamabe invariant by localizing the energy estimates near $\infty$. The estimate error is carefully corrected by keeping track of nonisotropic scaling. We bring the CR mass into the estimate explicitly, which provides us a global correction term in test functions estimate through the CR positive mass theorem.