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Complete densely embedded complex lines in $ \mathbb{C}^2$


Authors: Antonio Alarcón and Franc Forstnerič
Journal: Proc. Amer. Math. Soc. 146 (2018), 1059-1067
MSC (2010): Primary 32H02; Secondary 32E10, 32M17, 53A10
DOI: https://doi.org/10.1090/proc/13873
Published electronically: November 10, 2017
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Abstract: In this paper we construct a complete injective holomorphic immersion $ \mathbb{C}\to \mathbb{C}^2$ whose image is dense in $ \mathbb{C}^2$. The analogous result is obtained for any closed complex submanifold $ X\subset \mathbb{C}^n$ for $ n>1$ in place of $ \mathbb{C}\subset \mathbb{C}^2$. We also show that if $ X$ intersects the unit ball $ \mathbb{B}^n$ of $ \mathbb{C}^n$ and $ K$ is a connected compact subset of $ X\cap \mathbb{B}^n$, then there is a Runge domain $ \Omega \subset X$ containing $ K$ which admits a complete injective holomorphic immersion $ \Omega \to \mathbb{B}^n$ whose image is dense in $ \mathbb{B}^n$.


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Antonio Alarcón
Affiliation: Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spain
Email: alarcon@ugr.es

Franc Forstnerič
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana, Slovenia—and—Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia.
Email: franc.forstneric@fmf.uni-lj.si

DOI: https://doi.org/10.1090/proc/13873
Received by editor(s): February 25, 2017
Published electronically: November 10, 2017
Additional Notes: The first author was supported by the Ramón y Cajal program of the Spanish Ministry of Economy and Competitiveness and by the MINECO/FEDER grant No. MTM2014-52368-P, Spain.
The second author was partially supported by the research grants P1-0291 and J1-7256 from ARRS, Republic of Slovenia.
Communicated by: Filippo Bracci
Article copyright: © Copyright 2017 American Mathematical Society

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