The embedded Calabi-Yau conjecture for finite genus

TL;DR

This paper proves that certain minimal surfaces with finite genus and infinite ends have a limited number of simple limit ends, and characterizes their proper embedding in Euclidean space based on the number of limit ends.

Contribution

It establishes the embedded Calabi-Yau conjecture for finite genus minimal surfaces with infinite ends, linking the number of limit ends to properness.

Findings

Simple limit ends have properly embedded representatives with compact boundary and genus zero.

Surfaces with at least two simple limit ends have exactly two such ends.

Proper embedding is equivalent to having at most two limit ends.

Abstract

Suppose $M$ is a complete, embedded minimal surface in $\mathbb{R}^3$ with an infinite number of ends, finite genus and compact boundary. We prove that the simple limit ends of $M$ have properly embedded representatives with compact boundary, genus zero and with constrained geometry. We use this result to show that if $M$ has at least two simple limit ends, then $M$ has exactly two simple limit ends. Furthermore, we demonstrate that $M$ is properly embedded in $\mathbb{R}^3$ if and only if $M$ has at most two limit ends if and only if $M$ has a countable number of limit ends.