Holomorphic maps with large images

TL;DR

This paper constructs holomorphic maps from pseudoconvex domains to complex Euclidean spaces with large images, introducing a new concept of universal dominability and exploring boundary behaviors like Casorati-Weierstrass and Picard points.

Contribution

It introduces a novel approach to boundary value problems in complex analysis by constructing maps with controlled growth and boundary properties, and defines the concept of universal dominability.

Findings

Existence of holomorphic maps with large boundary images

Introduction of universal dominability concept

Construction of functions with prescribed boundary behaviors

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.