On the conformal and CR automorphism groups (Q1902007)

The author develops a new approach to study the behavior of conformal diffeomorphisms between Riemannian manifolds and CR diffeomorphisms between strictly pseudo-convex CR manifolds. It involves the scalar curvature theory and the conformally invariant Laplace operator and its analogue in the CR case. The following main results are proved. Theorem 1. The group \(G\) of conformal transformations of a Riemannian manifold \(M^n\) acts properly iff \(M^n\) is not conformally diffeomorphic to \(S^n\) or \(\mathbb{R}^n\) with the standard metric. Theorem 2. The group \(G\) of CR transformations of a strongly pseudoconvex CR manifold \(M^{2n+1}\) acts properly iff \(M^{2n + 1}\) is not CR diffeomorphic to the Heisenberg group \(H^{2n + 1}\) or the sphere \(S^{2n + 1}\) with the standard CR structure.