Global holomorphic extension of a local map and a Riemann mapping theorem for algebraic domains

Abstract

In his paper of 1907 [Po07], Poincare showed, among other things, that any local non-constant holomorphic map f from an open piece of the sphere ∂B into ∂B extends to a global biholomorphic map between the ball B. This result was generalized to B by Tanaka [Ta62] and was further established in a much more general setting by Alexander [A74]. Since then, the mixed equivalence problem was naturally formulated (see for instance [Wel82], [Be90]): How far can a local equivalence map between the boundaries of two nice domains be biholomorphically extended ? A significant contribution along the lines of this direction was made by Webster: In [We77], he not only extended Poincare’s theorem to real ellipsoids, but also did a pioneer work by showing a type of Chow’s theorem for local maps between algebraic domains. In particular, he showed that any local equivalence map between two smooth algebraic domains extends as a branched algebraic map. Webster’s algebraicity theorem was established in more general situations in the recent deep work of Baouendi, Ebenfelt and Rothschild (see [BR95], [BER96]), and in the first author’s work ([Hu94], [Hu94T]). In [CJ96], Chern and the second author showed a related result: Any local map f , which maps a piece of ∂D into ∂B, extends along any path γ ⊂ D as a locally bimeromorphic map, where D is a bounded domain with real analytic spherical boundary. The purpose of this paper is to study the global holomorphic extension of a local map between algebraic domains, i.e, domains whose boundaries are locally defined by real polynomials. Our first result is the following theorem: