Embedded minimal surfaces of finite topology

TL;DR

This paper characterizes complete embedded minimal surfaces with finite topology in three-dimensional space, showing they are conformally compact Riemann surfaces with finitely many punctures, and provides a classification of their asymptotic behavior.

Contribution

It proves that such minimal surfaces are of finite type and can be described via meromorphic data on their conformal compactification.

Findings

Minimal surfaces of finite topology are conformally compact Riemann surfaces.

These surfaces can be represented using meromorphic data.

The paper classifies the asymptotic behavior of these minimal surfaces.

Abstract

In this paper we prove that a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface $\overline{M}$ with boundary punctured in a finite number of interior points and that $M$ can be represented in terms of meromorphic data on its conformal completion $\overline{M}$. In particular, we demonstrate that $M$ is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of $M$.