Abstract

We prove that the domain $D$ in $\Gamma \times \mathbf{C}_z$ where $\Gamma$ is a polydisk centered at $(0)$ and the fiber of $D$ over every point of $\Gamma$ is a simply connected domain in $\mathbf{C}_z$ which contains a small disk $\{|z| \leqq \varepsilon \}$, where $\varepsilon$ is independent of every point of $\Gamma$, is biholomorophic to some complete Hartogs domain. And we give applications of the uniformization of some fiber spaces.

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