Complete minimal surfaces with Cantor ends in minimally convex domains

Abstract

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 $\Omega$ 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 $\Omega$.

arXiv commentTo appear in an issue of Pure Appl. Funct. Anal. dedicated to Nikolai Nadirashvili