Abstract. We prove a result on the structure of finite proper holomorphic mappings between complex manifolds that are products of hyperbolic Riemann surfaces. While an important special case of our result follows from the ideas developed by Remmert and Stein, the proof of the full result relies on the interplay of the latter ideas and a finiteness theorem for Riemann surfaces.

We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the pluricomplex Green's function. We also prove several other Schwarz lemmas using this function.In this section, we de ne and prove basic facts about an extremal function de ned using plurisubharmonic functions. Our treatment is from [Kob98, p. 184] where the de nition is attributed to Klimek [Kli85]. The paper by Demailly [Dem87] contains further properties of this function.De nition 4. Let X be a complex manifold. Fix z 0 ∈ X and de ne the extremal functionwhere P X (z 0 ) is the collection of functions ϕ on X that satisfy:1. ϕ is upper semi-continuous,

Abstract. We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in C n , n > 1. Specifically: we prove that any proper holomorphic selfmap of a certain type of balanced, finite-type domain in C n , n > 1, is an automorphism. The main novelty of our proof is the use of a recent result of Opshtein on the behaviour of the iterates of holomorphic self-maps of a certain class of domains. We use Opshtein's theorem, together with the tools made available by finiteness of type, to deduce that the aforementioned map is unbranched. The monodromy theorem then delivers the result.

The Minkowski function is a crucial tool used in the study of balanced domains and, more generally, quasi-balanced domains in several complex variables. If a quasi-balanced domain is bounded and pseudoconvex then it is well-known that its Minkowski function is plurisubharmonic. In this short note, we prove that under the additional assumption of smoothness of the boundary, the Minkowski function of a quasi-balanced domain is in fact smooth away from the origin. This allows us to construct a smooth plurisubharmonic defining function for such domains. Our result is new even in the case of balanced domains.2010 Mathematics Subject Classification. Primary 32A07.

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