2014/01/14 by Josip Globevnik, Globevnik, Josip
Mathematics · #32A40 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:32A40
paper · pdf · doi:10.48550/arxiv.1401.3135
23 pages
arxiv created 2014/01/14 · arxiv updated 2014/01/15
In 1977 P.Yang asked whether there exist complete immersed complex submanifolds g : Mk --> CN with bounded image. A positive answer is known for holomorphic curves (k=1) and partial answers are known for the case when k>1. The principal result of the present paper is a construction of a holomorphic function on the open unit ball BN of CN whose real part is unbounded on every path in BN of finite length that ends on the boundary of BN. A consequence is the existence of a complete, closed, complex hypersurface in BN. This gives a positive answer to Yang's question in all dimensions k, N, 1≤ k<N, by providing properly embedded complete complex manifolds.