  This paper is the fourth in a series where we describe the space of all
embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed
3-manifold. The key is to understand the structure of an embedded minimal disk
in a ball in $\RR^3$. This was undertaken in [CM3], [CM4] and the global
version of it will be completed here; see [CM15] for discussion of the local
case and [CM13], [CM14] where we have surveyed our results about embedded
minimal disks.
