Uniformization of strictly pseudoconvex domains. I

Abstract

It is shown that two strictly pseudoconvex Stein domains with real analytic boundaries have biholomorphic universal coverings provided that their boundaries are locally biholomorphically equivalent. This statement can be regarded as a higher dimensional analogue of the Riemann uniformization theorem.

On the Classification of Homogeneous Hypersurfaces in Complex Space

“…The domains D n t illustrate the relationship between the fundamental group of a smoothly bounded Stein domain and that of its (connected) boundary, which is important, for example, for the uniformization problem. Namely, if D is a smoothly bounded Stein domain, the fundamental groups π 1 (D) and π 1 (∂D) are isomorphic in complex dimensions ≥ 3, whereas in dimension 2 there exists a surjective homomorphism π 1 (∂D) → π 1 (D) and the fundamental group of ∂D can be larger than that of D (see [14] for a detailed discussion of these facts). Indeed, as we observed above,…”

“…Indeed, otherwise by results of [14] the universal cover of the domain D n t would be biholomorphic to the unit ball B n ⊂ C n . Since D n t is simply connected, this would imply that D n t is biholomorphic to B n , which is impossible since D n t is not contractible.…”

Higher Order Asymptotic Behavior of Certain Kähler Metrics and Uniformization for Strongly Pseudoconvex Domains

“…As a generalization of Chern-Ji's theorem ( [2]), Nemirovskii-Shaffikov proved in [11,12] that a strongly pseudoconvex domain Ω with C ∞ -smooth boundary is covered by the unit ball if every boundary point is spherical in the sense that all the CR invariants vanish identically on ∂Ω (cf. [3]).…”

Conjectures of Cheng and Ramadanov

“…It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.In this brief note we use our uniformization result from [10,11] to extend the work of Fu and Wong [7] on the relationship between two long-standing conjectures about the behaviour of the Bergman metric of a strictly pseudoconvex domain in C n , n ≥ 2. …”