Abstract
Suppose that 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 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 if M has at most two limit ends if and only if M has a countable number of limit ends.
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