Complete minimal surfaces with Cantor ends in minimally convex domains

We survey the recent history of the conformal Calabi-Yau problem consisting in determining the complex structures admitted by complete bounded minimal surfaces in ${\mathbb{R}}^3$. Moreover, we prove that for any minimally convex domain $\mathrm{\Omega }$ in ${\mathbb{R}}^3$ and any compact Riemann surface $R$ there is a Cantor set $C$ in $R$ whose complement $R\setminus C$ is the complex structure of a complete proper minimal surface in $\mathrm{\Omega }$.