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

2015/09/08 by Antonio Alarcon, Alarcon, Antonio, Josip Globevnik +3
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #math.CV #math.DG

paper · pdf · doi:10.48550/arxiv.1509.02283

20 pages, 3 figures. Main Theorem improved and proof simplified. To appear in J. Reine Angew. Math. (Crelle's J.)

arxiv created 2016/08/30 · arxiv updated 2016/08/31

Abstract

Given a closed complex hypersurface Z⊂ ℂ^N+1 (N∈ℕ) and a compact subset K⊂ Z, we prove the existence of a pseudoconvex Runge domain D in Z such that K⊂ 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 ℂ², with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc \mathbbD⊂ ℂ into the unit ball of ℂ². These are the first known examples of complete bounded embedded complex hypersurfaces in ℂ^N+1 with any control on the topology.

Related

• A complete complex hypersurface in the ball of C^N
• Complex curves in pseudoconvex Runge domains containing discrete subsets
• Complete complex hypersurfaces in the ball come in foliations
• On the existence of holomorphic embeddings of strictly pseudoconvex algebraic hypersurfaces into spheres
• Complete bounded embedded complex curves in C²