Embedded complex curves in the affine plane
Antonio Alarcon, Franc Forstneric

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 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 . The focal point is a lemma saying the following. Given a compact bordered Riemann surface, , a closed discrete subset of its interior , a compact subset without holes in , and a embedding which is holomorphic in , we can approximate uniformly on by a holomorphic embedding…
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Taxonomy
TopicsHolomorphic and Operator Theory · Meromorphic and Entire Functions · Analytic and geometric function theory
