Eric Bedford

Proper self maps of weakly pseudoconvex domains

This theorem was proved by Burns and Shnider I-5] in the case that D is strongly pseudoconvex. In the more general case, Pin6uk 1-9, Corollary 2] has reduced the problem of showing that a proper mapping f : D ~ D is biholomorphic to proving that f is unbranched, i.e., that det [ f ] + 0. Our interest in the theorem above stems from the fact that if f : D t ~ D 2 is a proper holomorphic mapping of a weakly pseudoconvex domain D 1 onto a second domain Dz, it is possible for f to branch if D1 +D 2. This is contrary to what may occur if D 1 is assumed to be strongly pseudoconvex (see Alexander [1], Pin6uk [-9, 10], Fornaess 1-8], Bell 1,3], Diederich and Fornaess 17]). It has recently been proved in 1-4] and in I-6] that the mapping f in the theorem extends smoothly to/) . This fact is basic to our proof. The theorem may be shown to remain valid if 112 is replaced by a Stein manifold (see [2]), but si~ace-,the arguments of [4] and [6] do not carry over automatically to this case, the methods used in 1-2] are more technical.

Domains of existence for plurisubharmonic functions

Theorem. Let 12 C ~ be a domain with C 2 boundary, and let the Levi form of OQ be nondeyenerate on an open dense subset of Og2. Then there is a plurisubharmonic function on £2 that cannot be extended to be plurisubharmonic on a set f2: CO with f21c~0:# 0. We note that the hypotheses of the theorem are generic and are satisfied, for instance, if £2 is bounded and has real analytic boundary. The methods here are local, but the situation where a large portion of ~£2 is Levi flat seems to pose a global problem. Let P(f2) denote the plurisubharmonic functions on O. It is well known that if g2 is pseudo-convex, then there is a plurisubharmonic exhaustion function for f2. However, if u~ Lipl(O)c~P(f2) and if f2 is strongly pseudoconvex with C 2 boundary, then u extends to a plurisubharmonic function in a neighborhood of O. On the other hand, if