We show that for any bounded domain Ω ⊂ C n of 1-type 2k which is locally convexifiable at p ∈ bΩ, having a Stein neighborhood basis, there is a biholomorphic map f :Ω → C n such that f (p) is a global extreme point of type 2k for f (Ω).

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For certain bordered submanifolds M ⊂ ℂ2 we show that M can be embedded properly and holomorphically into ℂ2. An application is that any subset of a torus with two boundary components can be embedded properly into ℂ2.

We prove that every circled domain in the Riemann sphere admits a proper holomorphic embedding into the affine plane ‫ރ‬ 2 .Theorem 1.1. Every domain in the Riemann sphere ‫ސ‬ 1 = ‫ރ‬ ∪ {∞} with at most countably many boundary components, none of which are points, admits a proper holomorphic embedding into ‫ރ‬ 2 .By the uniformization theorem of He and Schramm [1993], every domain in Theorem 1.1 is conformally equivalent to a circled domain, that is, a domain whose complement is a union of pairwise disjoint closed round discs.We prove the same embedding theorem also for generalized circled domains whose complementary components are discs and points (punctures), provided that all but finitely many of the punctures belong to the cluster set of the nonpoint boundary components (see Theorem 5.1). In particular, every domain in ‫ރ‬ or ‫ސ‬ 1 with at most countably many boundary components, at most finitely many of which are isolated points, admits a proper holomorphic embedding into ‫ރ‬ 2 (see Corollary 5.2 and Example 5.3).For finitely connected planar domains without isolated boundary points, Theorem 1.1 was proved by Globevnik and Stensønes [1995]. More recently it was shown by the authors in [Forstnerič and Wold 2009] that for every embedded complex curve C ⊂ ‫ރ‬ 2 , with smooth boundary bC consisting of finitely

We give necessary and sufficient conditions for totally real sets in Stein manifolds to admit Carleman approximation of class C k , k ≥ 1, by entire functions.