Uniformization of spherical CR manifolds

Abstract

Abstract. Let M be a closed (compact with no boundary) spherical CR manifold of dimension 2n + 1. Let M be the universal covering of M. Let Φ denote a CR developing mapwhere S 2n+1 is the standard unit sphere in complex n + 1-space C n+1 . Suppose that the CR Yamabe invariant of M is positive. Then we show that Φ is injective for n ≥ 3. In the case n = 2, we also show that Φ is injective under the condition: s(M ) < 1. It then follows that M is uniformizable.

A positive mass theorem in three dimensional Cauchy–Riemann geometry

“…It turns out that the term A is a multiple of the mass defined for the blow-up M. We observe that (10) holds for n = 1 (dimension 3 case) and for N being spherical of all dimensions. For such manifolds of dimension greater or equal to 5 (some extra technical condition in dimension 5) with positive CR Yamabe or Tanaka-Webster class, one can prove a positive mass theorem for A (and hence find solutions of the CR Yamabe problem with minimal energy) through another approach ( [18]).…”

Convergence of the CR Yamabe flow

“…When M is spherical, give any x ∈ M , we can find a smooth function ϕ x such that θ H n = ϕ Under the assumptions of Theorem 1.1, the CR mass is positive, i.e. A x > 0 for all x ∈ M by the CR positive mass theorem (see Theorem 1.1 in [11] and Corollary C in [10]). Note that the function x → A x is continuous.…”

“…For every 1 ≤ k ≤ m, we define u (x k ,ε k ) by (A. 10) u (x k ,ε k ) (y) = ϕ x k (y) U (x k ,ε k ) (y).…”

A generalization of the Greene–Krantz theorem for the semicontinuity property of automorphism groups

“…Let G k be the Green function for L θ k with pole at y k . Existence of the Green function follows from the fact that Y k > 0 (see for instance, [4,9]). We normalize G k by the condition min M\{y k } G k = 1.…”

“…The main tool of the proof is the solution for the CR Yamabe problem about the construction of pseudohermitian structures with constant Webster scalar curvature, which is intensively studied for instance, in [4,9,10,18,19]. The subellipticity of the CR Yamabe equation turned out quite useful in obtaining estimates of derivatives of CR automorphisms by Schoen in [27].…”