The global theory of complete minimal surfaces in \({\mathbb R}^3\) has been developed for almost two and one-half centuries. One of the central problems in this theory is the so-called Calabi-Yau problem, which is studied in this paper. In 1996, N. Nadirashvili proved the following theorem [Invent. Math. 126, 457–465 (1996; Zbl 0881.53053)]. There exists a complete minimal immersion \(f:{\mathbb D}\to {\mathbb B}\) from the open unit disk \(\mathbb D\) into the open unit ball \({\mathbb B}\subset {\mathbb R}^3 \). Furthermore, the immersion can be constructed with negative Gaussian curvature. However, the proof of Nadirashvili did not guarantee that the immersion \(f\) was proper, where by proper we mean that \(f^{-1}(C)\) is compact for any \(C\subset {\mathbb B}\) compact.
In this paper the authors answer the question of the existence of complete, simply connected minimal surfaces that are proper in convex regions of \({\mathbb R}^3\). The main theorems of this paper are as follows: If \({\mathbb B}\subset{\mathbb R}^3\) is a convex domain (not necessarily bounded or smooth), then there exists a complete, proper minimal immersion \(\psi :{\mathbb D}\to{\mathbb B}\). Let \(D'\) be a convex domain of space, and consider a bounded convex regular domain \(D\) with \({\overline{D}}\subset {D'}\). Then any minimal disk whose boundary lies in \(\partial D\) can be approximated in any compact subdomain of \(D\) by a complete minimal disk that is proper in \(D'\).