Embedding some Riemann surfaces into $${\mathbb {C}^2}$$ with interpolation

Kutzschebauch, Frank; Løw, Erik; Wold, Erlend Fornaess

doi:10.1007/s00209-008-0392-8

Cited by 19 publications

(18 citation statements)

References 18 publications

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“…The main result of this section is the following theorem, proved by Wold for C 2 in [37,38], and with interpolation by Kutzschebauch, Løw and Wold in [22]. Theorem 1 (Wold embedding theorem for C × C * ).…”

Section: Definitionmentioning

confidence: 91%

“…In Section 3 we will show that the embedding σ so constructed is homotopic to ψ| X . In proving Theorem 1 we will follow the argument given in [22], making a number of modifications so that the proof holds with target space C × C * for the embedding, rather than C 2 . The proof will require a number of preliminary results and definitions, and we begin by reminding the reader what it means for a vector field to be R-complete.…”

Section: Definitionmentioning

confidence: 99%

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A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

Ritter

2011

J Geom Anal

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Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into C× C * , namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalisation to C × C * of recent results of Wold and Forstnerič on the long-standing problem of properly embedding open Riemann surfaces into C 2 , with an additional result on the homotopy class of the embeddings. We also give a complete solution to a question that arises naturally in Lárusson's holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.

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Section: Definitionmentioning

confidence: 91%

Section: Definitionmentioning

confidence: 99%

A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

Ritter

2011

J Geom Anal

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“…Since K := B N ∪ (X ∩ B N +1 ) is polynomially convex there is an open neighborhood Ω of K such that Ω is polynomially convex and such that Ω ∩ ϕ(Γ) = ∅. Thus by Lemma 2.3 in [5] there exists a Φ ∈ Aut hol (C 2 ) such that Φ − Id Ω < ε and such that Φ(ϕ(Γ)) ⊂ C 2 \ B N +2 . By possibly having to decrease ε we may assume that Φ(ϕ(W )) ∩ (X ∩ B N +1 ) = ∅ and so we may putφ := Φ • ϕ.…”

Section: Constructionmentioning

confidence: 96%

Proper holomorphic disks in the complement of varieties in $\mathbb{C}^2$

Borell

Kutzschebauch

Wold

2008

Mathematical Research Letters

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“…The nice projection property for a finite collection of smoothly embedded curves in C × C * was given in [9], based on a definition introduced in [8]. The definition in [9] concerns properties of the curves after they have been projected onto the C-component 6 of C × C * using π 1 , and in this paper we refer to it as the C-nice projection property.…”

Section: Definitionmentioning

confidence: 99%

“…The current paper, like the second author's earlier paper [9], relies on the fundamental embedding technique introduced by Wold in [11]. Later papers by Forstnerič and Wold [4], and Kutzschebauch, Løw, and Wold [8] have also been of use to us.…”

Section: Introductionmentioning

confidence: 99%

Proper holomorphic immersions in homotopy classes of maps from finitely connected planar domains into CxC*

Lárusson¹,

Ritter²

2014

Indiana Univ. Math. J.

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Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. Previously we have shown that, given a continuous map X → C×C * from a finitely connected planar domain X without isolated boundary points, a stronger Oka property holds, namely that the map is homotopic to a proper holomorphic embedding. Here we show that every continuous map from a finitely connected planar domain, possibly with punctures, into C × C * is homotopic to a proper immersion that identifies at most countably many pairs of distinct points, and in most cases, only finitely many pairs. By examining situations in which the immersion is injective, we obtain a strong Oka property for embeddings of some classes of planar domains with isolated boundary points. It is not yet clear whether a strong Oka property for embeddings holds in general when the domain has isolated boundary points. We conclude with some observations on the existence of a null-homotopic proper holomorphic embedding C * → C × C * .

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Embedding some Riemann surfaces into $${\mathbb {C}^2}$$ with interpolation

Abstract: We prove that several types of open Riemann surfaces, including the finitely connected planar domains, embed properly into C 2 such that the values on any given discrete sequence can be arbitrarily prescribed. Mathematics Subject Classification (2000)

Cited by 19 publications

References 18 publications

A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

Proper holomorphic disks in the complement of varieties in $\mathbb{C}^2$

Proper holomorphic immersions in homotopy classes of maps from finitely connected planar domains into CxC*

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