2001/01/31 by Franc Forstneric, Josip Globevnik
Mathematics · #math.CV #msc:32H02 #msc:32H35 #msc:32E10 #msc:32E35
published as Math. Res. Lett. 8, 257-274 (2001) · Math. Res. Lett. (to appear)
arxiv created 2001/02/27 · arxiv updated 2009/11/30
In this paper we investigate the global behavior of proper holomorphic maps f from the unit disc U=|z|<1 to C2. The fact that U is transcendental imposes certain restrictions on the image f(U). For instance, f(U) cannot be contained in any proper complex cone in C2 since this would force it to be algebraic. On the other hand, we show that a real cone in C2 with axis R2 contains the image of a proper holomorphic map f from U to C2 if and only if the angle of the cone is larger than pi/2. We also construct maps f as above whose images avoid both coordinate axes in C2. Equivalently, we construct a pair of positive harmonic functions u, v on U such that maxu(z),v(z) tends to plus infinity when z tends to the boundary of U. Furthermore we show that the components f1, f1 of a proper holomorphic map from U to C2, as well as polynomial and certain rational functions of f1 and f2, have the property that their essential range at any boundary point of U omits at most a polar set in C.