Complete proper minimal surfaces in convex bodies of $\r^3$ (II): The behavior of the limit set

Details Complete proper minimal surfaces in convex bodies of $\r^3$ (II): The behavior of the limit set Complete proper minimal surfaces in convex bodies of $\r^3$ (II): The behavior of the limit set

2005-05-24

arxiv.org/abs/math/0505501v1

Mathematics

Differential Geometry

21 pages, 5 figures

Scientific paper

Let $D$ be a regular strictly convex bounded domain of $\r^3$, and consider a regular Jordan curve $\Gamma \subset \partial D$. Then, for each $\epsilon>0$, we obtain the existence of a complete proper minimal immersion $\psi_\epsilon :\d \to D$ satisfying that the Hausdorff distance $\delta^H(\psi_\epsilon(\partial \d), \Gamma) < \epsilon,$ where $\psi_\epsilon(\partial \d)$ represents the limit set of the minimal disk $\psi_\epsilon(\d).$ This result has some interesting consequences. Among other things, we can prove that any bounded regular domain $R$ in $\r^3$ admits a complete proper minimal immersion $\psi: \d \longrightarrow R$.

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