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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


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Volume 2019, Issue 751

Issues

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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/ 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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/ 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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Published Online: 2016-11-08 | DOI: https://doi.org/10.1515/crelle-2016-0061

Abstract

Given a closed complex hypersurface ZN+1 (N) and a compact subset KZ, we prove the existence of a pseudoconvex Runge domain D in Z such that KD and there is a complete proper holomorphic embedding from D into the unit ball of N+1. For N=1, we derive the existence of complete properly embedded complex curves in the unit ball of 2, with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc 𝔻 into the unit ball of 2. These are the first known examples of complete bounded embedded complex hypersurfaces in 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, DOI: https://doi.org/10.1515/crelle-2016-0061.

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