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

作者：Ritter Tyson^*

来源：Journal of Geometric Analysis, 2013, 23(2): 571-597.

DOI：10.1007/s12220-011-9254-4

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

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 a%26quot;,xa%26quot;,(au), namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalization to a%26quot;,xa%26quot;,(au) of recent results of Wold and Forstneri on the long-standing problem of properly embedding open Riemann surfaces into a%26quot;,(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 Larusson%26apos;s holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.

• 出版日期2013-4