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

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

2003-01-13

arxiv.org/abs/math/0301132v1

Mathematics

Differential Geometry

23 pages, 4 figures

Scientific paper

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.

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