Complete minimal surfaces with Cantor ends in minimally convex domains

TL;DR

This paper surveys the conformal Calabi-Yau problem and proves the existence of complete proper minimal surfaces with Cantor ends in minimally convex domains, expanding understanding of minimal surface structures.

Contribution

It introduces the construction of complete minimal surfaces with Cantor set ends in minimally convex domains, advancing the theory of minimal surfaces in complex analysis.

Findings

Existence of complete proper minimal surfaces with Cantor ends in minimally convex domains.

Connection between complex structures on Riemann surfaces and minimal surfaces.

Progress in solving the conformal Calabi-Yau problem.

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 $\Omega$.