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 $|f|\le C_{\alpha }{e}^{C_{\alpha }^{\prime }{\hat{\delta }}^{-\alpha }}$, 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.

A supplement on the definition of Picard points for holomorphic maps was added at the end of the paper