On the existence of a proper minimal surface in $R^3$ with the conformal type of a disk

Abstract

The main goal of this paper is to show a counterexample to the following conjecture: {\bf Conjecture} [Meeks, Sullivan]: If $f:M\to \mathbb{R}^3$ is a complete proper minimal immersion where $M$ is a Riemannian surface without boundary and with finite genus, then $M$ is parabolic. We have proved: {\bf Theorem:} There exists $χ: D\longrightarrow \mathbb{R}^3$, a conformal proper minimal immersion defined on the unit disk.