Journal of Geometric Analysis

, Volume 23, Issue 2, pp 571–597 | Cite as

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

  • Tyson RitterEmail author


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 generalization to ℂ×ℂ of recent results of Wold and Forstnerič on the long-standing problem of properly embedding open Riemann surfaces into ℂ2, 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.


Holomorphic embedding Riemann surface Oka principle Stein manifold Elliptic manifold Acyclic map Circular domain Fatou–Bieberbach domain 

Mathematics Subject Classification (2000)

32Q40 32E10 32H02 32H35 32M17 32M25 32Q28 


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© Mathematica Josephina, Inc. 2011

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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