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

Abstract

Let $D$ be a regular strictly convex bounded domain of ${\mathbb{R}}^3$, 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 ${\psi }_{\epsilon }:\mathbb{D}\to D$ satisfying that the Hausdorff distance ${\delta }^H({\psi }_{\epsilon }(\partial \mathbb{D}),\Gamma )<\epsilon ,$ where ${\psi }_{\epsilon }(\partial \mathbb{D})$ represents the limit set of the minimal disk ${\psi }_{\epsilon }(\mathbb{D}).$ This result has some interesting consequences. Among other things, we can prove that any bounded regular domain $R$ in ${\mathbb{R}}^3$ admits a complete proper minimal immersion $\psi :\mathbb{D}⟶R$.

21 pages, 5 figures