Holomorphic functions unbounded on curves of finite length

TL;DR

This paper constructs holomorphic functions on pseudoconvex domains in complex space that are unbounded on certain finite-length curves, leading to the existence of complete complex hypersurfaces with similar properties.

Contribution

It proves the existence of holomorphic functions with unbounded behavior on finite-length paths in pseudoconvex domains, and constructs complete complex hypersurfaces with this property.

Findings

Existence of holomorphic functions unbounded on finite-length paths

Construction of complete complex hypersurfaces with unbounded path lengths

Implications for the geometry of pseudoconvex domains

Abstract

Given a pseudoconvex domain D in C^N, N>1, we prove that there is a holomorphic function f on D such that the lengths of paths p: [0,1]--> D along which Re f is bounded above, with p(0) fixed, grow arbitrarily fast as p(1)--> bD. A consequence is the existence of a complete closed complex hypersurface M in D such that the lengths of paths p:[0,1]--> M, with p(0) fixed, grow arbitrarily fast as p(1)-->bD.