The embedded Calabi-Yau conjecture for finite genus

Abstract

Suppose M is a complete, embedded minimal surface in R3 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 R3 if and only ifM has at most two limit ends if and only ifM has a countable number of limit ends.