Holomorphic Maps with Large Images

Abstract

We show that each pseudoconvex domain \(\Omega \subset {\mathbb {C}}^n\) admits a holomorphic map \(F\) to \({\mathbb {C}}^m\) with \(|F|\le C_1 e^{C_2 \hat{\delta }^{-6}}\), where \(\hat{\delta }\) is the minimum of the boundary distance and \((1+|z|^2)^{-1/2}\), such that every boundary point is a Casorati–Weierstrass point of \(F\). Based on this fact, we introduce a new anti-hyperbolic concept—universal dominability. We also show that for each \(\alpha >6\) and each pseudoconvex domain \(\Omega \subset {\mathbb C}^n\), there is a holomorphic function \(f\) on \({\Omega }\) with \(|f|\le C_\alpha e^{C_\alpha ' \hat{\delta }^{-\alpha }}\), such that every boundary point is a Picard point of \(F\). Applications to the construction of holomorphic maps of a given domain onto some \({\mathbb C}^m\) are given.