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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

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1435-5345
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Volume 2019, Issue 751

# A construction of complete complex hypersurfaces in the ball with control on the topology

Antonio Alarcón
• Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spain
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Josip Globevnik
• Department of Mathematics, University of Ljubljana; and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Francisco J. López
• Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spain
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
Published Online: 2016-11-08 | DOI: https://doi.org/10.1515/crelle-2016-0061

## Abstract

Given a closed complex hypersurface $Z\subset {ℂ}^{N+1}$ ($N\in ℕ$) and a compact subset $K\subset Z$, we prove the existence of a pseudoconvex Runge domain D in Z such that $K\subset D$ and there is a complete proper holomorphic embedding from D into the unit ball of ${ℂ}^{N+1}$. For $N=1$, we derive the existence of complete properly embedded complex curves in the unit ball of ${ℂ}^{2}$, with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc $𝔻\subset ℂ$ into the unit ball of ${ℂ}^{2}$. These are the first known examples of complete bounded embedded complex hypersurfaces in ${ℂ}^{N+1}$ with any control on the topology.

## References

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## About the article

Received: 2016-03-25

Revised: 2016-05-04

Published Online: 2016-11-08

Published in Print: 2019-06-01

Antonio Alarcón is supported by the Ramón y Cajal program of the Spanish Ministry of Economy and Competitiveness. Antonio Alarcón and Francisco J. López are partially supported by the MINECO/FEDER grant no. MTM2014-52368-P, Spain. Josip Globevnik is partially supported by the research program P1-0291 and the grant J1-5432 from ARRS, Republic of Slovenia.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 751, Pages 289–308, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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