Consider a convex domain of . We prove that there exist complete minimal surfaces that are properly immersed in . We also demonstrate that if and are convex domains with bounded and the closure of contained in , then any minimal disk whose boundary lies in the boundary of can be approximated in any compact subdomain of by a complete minimal disk that is proper in . 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 is not a universal region for minimal surfaces, in the sense explained by Meeks and Pérez in .
"Complete proper minimal surfaces in convex bodies of ." Duke Math. J. 128 (3) 559 - 593, 15 June 2005. https://doi.org/10.1215/S0012-7094-04-12835-0