Extension of biholomorphic maps from special domains into their boundaries -

Abstract

Several theorems that hold in the theory of one complex variable cannot be generalized to the theory of several complex variables. One of them is the Riemann Mapping Theorem, which states that every non-empty simply connected domain which is not the entire complex plane is biholomorphic to the open unit disc, and from which follows the fact that any two non-empty proper simply-connected domains in C are biholomorphic. In this paper, we show that the last statement is not true in Cn for n ≥ 2 using the properties of the Levi form. However, the proof entails, assuming the existence of such a biholomorphism f, some boundary information about f. This requirement is fulfilled alluding to Fefferman Theorem, which will be proved for special domains using powerful tools: extremal and stationary maps.