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

TL;DR

This paper constructs a counterexample to a conjecture by demonstrating a proper minimal surface in three-dimensional space with the conformal type of a disk, challenging previous assumptions about such surfaces.

Contribution

It provides the first known example of a proper minimal surface in R^3 conformally equivalent to a disk, countering the conjecture that all such surfaces are parabolic.

Findings

Existence of a proper minimal surface with disk conformal type.

Counterexample to Meeks and Sullivan's conjecture.

Challenges previous beliefs about the conformal types of minimal surfaces.

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 $\chi: D\longrightarrow \mathbb{R}^3$, a conformal proper minimal immersion defined on the unit disk.