arXiv 1702.080322017-02math.CVmath.DG

Complete densely embedded complex lines in $\mathbb{C}^2$

Antonio Alarcon, Franc Forstneric

备注：: To appear in Proc. Amer. Math. Soc
发表信息：: Proc. Amer. Math. Soc. 146:3 (2018) 1059-1067
DOI：: 10.1090/proc/13873

英文摘要：

In this paper we construct a complete injective holomorphic immersion $\mathbb{C}\to\mathbb{C}^2$ whose image is dense in $\mathbb{C}^2$. The analogous result is obtained for any closed complex submanifold $X\subset \mathbb{C}^n$ for $n>1$ in place of $\mathbb{C}\subset\mathbb{C}^2$. We also show that, if $X$ intersects the unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ and $K$ is a connected compact subset of $X\cap\mathbb{B}^n$, then there is a Runge domain $Ω\subset X$ containing $K$ which admits a complete holomorphic embedding $Ω\to\mathbb{B}^n$ whose image is dense in $\mathbb{B}^n$.

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