The embedded Calabi-Yau conjecture for finite genus

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.