Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings

Abstract

A classical theorem of Carath6odory [8] states that every biholomorphic map f: D~ ~D2 between domains in the complex plane C bounded by simple closed Jordan curves extends to a homeomorphism of/31 onto/ )2 . There are some well-known generalizations of this result to domains in IE a. If D t and D2 are bounded pseudoconvex domains in 112" with if2 boundary and if the infinitezimal Kobayashi metric on D2 grows sufficiently fast near the boundary of D2(Ko2(g; X) >= [Xl/d(z, bDz) ~ for some e e (0, 1)), then every proper holomorphic map f : Dt--*D2 extends to a H61der-continuous map of/51 onto/32 [2, 3, 10, 25, 29-32]. This holds in particular if/92 is strictly pseudoconvex or if it is pseudoconvex with real-analytic boundary. The same result holds if D2 is only piecewise smooth strongly pseudoconvex [31]. Further results treat exceptional cases such as the balls [1 ], Reinhardt domains [5, 24], domains with many symmetries [4], and analytically bounded Hartogs domains in C 2 [I1]. Biholomorphic maps between certain types of nonpseudoconvex domains with real-analytic boundaries were treated in [13] and [27]. Besides the papers mentioned above there is a vast literature concerning smooth extension of proper holomorphic maps of smoothly bounded pseudoconvex domains ([6, 7, 12, 14], to mention just a few). In the present paper we obtain some results on the continuous extension of proper holomorphic maps f : D1--*/)2 under local assumptions on the boundaries bD1 and bD2. One of the main results is