Embedded complex curves in the affine plane

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 ${\mathcal{C}}^1$ 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.

arXiv commentThe new Theorems 1.7 and 1.9 and Section 8 have been added

References in corpus (9)

• Bordered Riemann surfaces in C^2
• Embeddings of infinitely connected planar domains into C^2
• Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
• A Carleman type theorem for proper holomorphic embeddings
• Complete embedded complex curves in the ball of ${\mathbb{C}}^2$ can have any topology
• An interpolation theorem for proper holomorphic embeddings
• Minimal surfaces in minimally convex domains
• Complete densely embedded complex lines in ${\mathbb{C}}^2$
• The Calabi-Yau problem for minimal surfaces with Cantor ends

Cited by in corpus (2)

• Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure
• Oka-1 manifolds