The embedded Calabi–Yau conjecture for finite genus

Abstract

Suppose that 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 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.