Complete Proper Minimal Surfaces in Convex Bodies of R

Consider a convex domain B of R 3. We prove that there exist complete minimal surfaces which are properly immersed in B. We also demonstrate that if D and D ′ are convex domains with D bounded and the closure of D contained in D ′ 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 which is proper in D ′. 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].