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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 3manifold. The key is to understand the structure of an embedded minimal disk in a ball in R3 . This was undertaken in [CM3], [CM4] and the global version of it will be completed here; see the discussion around Figure 12 for the local case and [CM15] for some more details. Our main results are Theorem 0.1 (the lamination theorem) and Theorem 0.2 (the one-sided curvature estimate). . | Annals of Mathematics The space of embedded minimal surfaces of fixed genus in a 3-manifold IV Locally simply connected By Tobias H. Colding and William P. Minicozzi II Annals of Mathematics 160 2004 573 615 The space of embedded minimal surfaces of fixed genus in a 3-manifold IV Locally simply connected By Tobias H. Colding and William P. Minicozzi II 0. Introduction 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 R3. This was undertaken in CM3 CM4 and the global version of it will be completed here see the discussion around Figure 12 for the local case and CM15 for some more details. Our main results are Theorem 0.1 the lamination theorem and Theorem 0.2 the one-sided curvature estimate . The lamination theorem is stated in the global case where the lamination is in fact a foliation. The first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph. This is done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase like the helicoid. To prove that such a disk is a double spiral staircase we showed in the first three papers of the series that it is built out of N-valued graphs where N is a fixed number. In this paper we will deal with how the multi-valued graphs fit together and in particular prove regularity of the set of points of large curvature - the axis of the double spiral staircase. The first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase. Theorem 0.1 see Figure 1 . Let Lị c BRi BRi 0 c R3 be a sequence of embedded minimal disks with dLi c dBRi where Ri TO. If sup A 2 TO BiOSi The first author was partially supported by
