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 $Γ\subset \partial D$. Then, for each $ε>0$, we obtain the existence of a complete proper minimal immersion $ψ_ε:\mathbb{D} \to D$ satisfying that the Hausdorff distance $δ^H(ψ_ε(\partial \mathbb{D}), Γ) < ε,$ where $ψ_ε(\partial \mathbb{D})$ represents the limit set of the minimal disk $ψ_ε(\mathbb{D}).$ 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 $ψ: \mathbb{D} \longrightarrow R$.