Embeddings of infinitely connected planar domains into ℂ²

Abstract

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

Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey

“…The second result due to Wold and the author [61] (2013) concerns domains with infinitely many boundary components. A domain X in the Riemann sphere P 1 is a generalized circled domain if every connected component of P 1 \ X is a round disc or a point.…”

Complex Geometry and Dynamics

“…By introducing the technique of exposing boundary points alluded to above, combined with the Andersén-Lempert theory of holomorphic automorphisms of C n for n > 1, Forstnerič and Wold proved in 2009 that, if a compact bordered Riemann surface M admits a (nonproper) holomorphic embedding in C 2 then its interior M admits a proper holomorphic embedding in C 2 [48]. Further applications of their technique can be found in [69] and [49]. For example, every circular domain in the Riemann sphere admits a proper holomorphic embedding in C 2 [49].…”

“…Further applications of their technique can be found in [69] and [49]. For example, every circular domain in the Riemann sphere admits a proper holomorphic embedding in C 2 [49]. (The case of finitely connected plane domains was established by Globevnik and Stensønes in 1995 [55].)…”

“…(The case of finitely connected plane domains was established by Globevnik and Stensønes in 1995 [55].) The main novelty in [48,49] is that, unlike in the earlier constructions, no cutting of the surface is needed and hence the conformal structure is preserved. It is considerably easier to show that every open oriented real surface M admits a complex structure J such that the open Riemann surface (M, J) admits a proper holomorphic embedding into C 2 (Alarcón and López [11]; for the case of finite topology see [31]).…”

Complete embedded complex curves in the ball of ℂ2 can have any topology

“…The latter contributes to the so-called embedding problem for open Riemann surfaces in C 2 ; a long-standing open question in Riemann Surface Theory asking whether every open Riemann surface properly embeds in C 2 as a complex curve (cf. Bell and Narasimhan [7, Conjecture 3.7, page 20]; for recent advances and a history of this classical problem we refer to the works by Forstnerič and Wold [12,13] and the references therein). It is shown in [12] that given a compact bordered Riemann surface M = M ∪ bM admitting a smooth embedding f : M ֒→ C 2 which is holomorphic in M , there is a proper holomorphic embedding f : M ֒→ C 2 which is as close as desired to f uniformly on a given compact subset of M .…”