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

• Antonio Alarcón , Josip Globevnik and Francisco J. López

## 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.

Funding statement: 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.

## Acknowledgements

The authors wish to thank Jean-Pierre Demailly for suggesting the use of tangent balls instead of faces of convex polytopes. They are also grateful to the referee for very useful comments that led to improvement of the original manuscript.

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