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 ' 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 that 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].
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