Suppose that M is a complete, embedded minimal surface in 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 if and only if M has at most two limit ends if and only if M has a countable number of limit ends.
"The embedded Calabi–Yau conjecture for finite genus." Duke Math. J. 170 (13) 2891 - 2956, 15 September 2021. https://doi.org/10.1215/00127094-2020-0087