Complete proper minimal surfaces in convex bodies of ℝ 3

Abstract

Consider a convex domain $B$ of ${ℝ}^3$. We prove that there exist complete minimal surfaces that are properly immersed in $B$. We also demonstrate that if $D$ and $D\text{'}$ are convex domains with $D$ bounded and the closure of $D$ contained in $D\text{'}$, then any minimal disk whose boundary lies in the boundary of $D$ can be approximated in any compact subdomain of $D$ by a complete minimal disk that is proper in $D\text{'}$. We apply these results to study the so-called type problem for a minimal surface: we demonstrate that the interior of any convex region of ${ℝ}^3$ is not a universal region for minimal surfaces, in the sense explained by Meeks and Pérez in [9].