Finite topology minimal surfaces in homogeneous three-manifolds

We prove that any complete, embedded minimal surface M with finite topology in a homogeneous three-manifold N has positive infectivity radius. When one relaxes the condition that N be homogeneous to that of being locally homogeneous, then we show that the closure of M has the structure of a minimal lamination of N. As an application of this general result we prove that any complete, embedded minimal surface with finite genus and a countable number of ends is compact when the ambient space is S-3 equipped with a homogeneous metric of nonnegative scalar curvature. (C) 2017 Elsevier Inc. All rights reserved.