Structure theorems for singular minimal laminations

In this paper, we apply our local removable singularity theorem and local struc-ture theorems for embedded minimal surfaces and minimal laminations in R3 proven in [16, 15], to obtain global structure theorems for certain possibly singular minimal laminations of R3. We will use Theorems 1.3 and Theorem 1.6 below in [14] to prove that a complete, embedded minimal surface in R3 with finite genus and a countable number of ends is proper. Theorem 1.6 will also be applied in [13] to obtain bounds on the index and the topology of complete, embedded minimal surfaces of fixed genus and finite topology in R3