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

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- by Antonio Alarcón and Franc Forstnerič PDF
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**146**(2018), 1059-1067 Request permission

## 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 $\mathbb {B}^n$.## References

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## Additional Information

**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
- MR Author ID: 783655
- 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.
- MR Author ID: 228404
- Email: franc.forstneric@fmf.uni-lj.si
- 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
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3750218