Embedded complex curves in the affine plane

TL;DR

This paper advances the understanding of embedding open Riemann surfaces into the affine plane, providing new constructions and approximation techniques that address longstanding conjectures and problems in complex geometry.

Contribution

It introduces a key lemma enabling approximation and interpolation of holomorphic embeddings of bordered Riemann surfaces into C^2, contributing to solutions of classical conjectures.

Findings

Existence of proper holomorphic embeddings with interpolation in C^2.

Every compact Riemann surface's complement of a Cantor set admits a proper embedding.

A new approximation lemma for holomorphic embeddings of bordered Riemann surfaces.

Abstract

This paper brings several contributions to the classical Forster-Bell-Narasimhan conjecture and the Yang problem concerning the existence of proper and almost proper (hence complete) injective holomorphic immersions of open Riemann surfaces in the affine plane $\mathbb C^2$ satisfying interpolation and hitting conditions. We also show that in every compact Riemann surface there is a Cantor set whose complement admits a proper holomorphic embedding in $\mathbb C^2$. The focal point is a lemma saying the following. Given a compact bordered Riemann surface, $M$, a closed discrete subset $E$ of its interior $\mathring M=M\setminus bM$, a compact subset $K\subset \mathring M\setminus E$ without holes in $\mathring M$, and a $\mathscr C^1$ embedding $f:M\hookrightarrow \mathbb C^2$ which is holomorphic in $\mathring M$, we can approximate $f$ uniformly on $K$ by a holomorphic embedding $F:M\hookrightarrow \mathbb C^2$ which maps $E\cup bM$ out of a given ball and satisfies some interpolation conditions.