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

TL;DR

This paper extends the Oka principle to show that continuous maps from finitely connected planar domains into * can be homotoped to proper holomorphic immersions, often with finitely many double points, and explores conditions for embeddings.

Contribution

It demonstrates that such maps can be approximated by proper holomorphic immersions with controlled double points, advancing understanding of holomorphic embeddings in complex geometry.

Findings

Every continuous map from a finitely connected planar domain into * is homotopic to a proper immersion.

Most cases allow the immersion to identify only finitely many pairs of points.

Conditions are identified under which the immersion can be an embedding, especially for domains with isolated boundary points.

Abstract

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 $X \to \C\times\C^*$ 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 $\C\times\C^*$ 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 $\C^* \to \C\times\C^*$.