Embeddings of infinitely connected planar domains into C^2

TL;DR

This paper proves that all circled domains in the Riemann sphere can be properly embedded into C^2, extending to punctured domains under certain conditions, using novel holomorphic embedding techniques.

Contribution

It introduces new methods for embedding circled domains and punctured domains into C^2, broadening the class of domains known to admit such embeddings.

Findings

Every circled domain in the Riemann sphere admits a proper holomorphic embedding into C^2.

The methods extend to certain punctured domains with finitely many punctures in the closure of complementary discs.

The results generalize previous embedding theorems for planar domains.

Abstract

We prove that every circled domain in the Riemann sphere admits a proper holomorphic embedding to C^2. Our methods also apply to circled domains with punctures, provided that all but finitely many of the punctures belong to the closure of the set of complementary discs.