Bordered Riemann surfaces in C^2

TL;DR

This paper proves that certain bordered Riemann surfaces with specific holomorphic immersions can be properly embedded into C^2, advancing the understanding of the classical problem of embedding open Riemann surfaces.

Contribution

It establishes a new sufficient condition for proper holomorphic embedding of bordered Riemann surfaces into C^2, providing an explicit construction without Teichmuller space theory.

Findings

Provides an explicit elementary construction for embeddings

Extends previous results by Globevnik, Stensones, and Wold

Offers an alternative proof approach using Teichmuller theory

Abstract

One of the oldest open problems in the classical function theory is whether every open Riemann surface admits a proper holomorphic embedding into C^2. In this paper we prove the following Theorem: If D is a bordered Riemann surface whose closure admits an injective immersion in C^2 that is holomorphic in D, then D admits a proper holomorphic embedding in C^2. The most general earlier results are due to J. Globevnik and B. Stensones (Math. Ann. 303 (1995), 579-597) and E. F. Wold (Internat. J. Math. 17 (2006), 963-974). We give an explicit and elementary construction that does not require the Teichmuller space theory, and we also indicate another possible proof using the latter theory.