The Space of Embedded Minimal Surfaces of Fixed Genus in a 3-manifold V; Fixed Genus

This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for nonsimply connected embedded minimal surfaces of any given fixed genus. The first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely–oriented double spiral staircases; see Figure 1. The second gives a pair of pants decomposition of any such surface when there are small necks, cutting the surface along a collection of short curves; see Figure 2. After the cutting, we are left with graphical pieces that are defined over a disk with either one or two sub–disks removed (a topological disk with two sub–disks removed is called a pair of pants). Both of these structures occur as different extremes in the two-parameter family of minimal surfaces known as the Riemann examples. The results of [CM3]–[CM6] have already been used by many authors; see, e.g., the surveys [MeP], [P], [Ro] and the introduction in [CM6] for some of these applications. There is much current research on minimal surfaces with infinite topology. Some of the results of the present paper were announced previously and have already been widely used to study infinite topology minimal surfaces; see, e.g., [MeP], [MePRs1], [MePRs2], [MePRs3], and [P]. The two main structure theorems for non-simply connected surfaces: Figure 1. Absence of necks: The surface can be obtained by gluing together two oppositely–oriented double spiral staircases. The curves that we cut along. One of the “pair of pants” (in bold). Figure 2. Presence of necks: The surface can be decomposed into a collection of pair of pants by cutting along short curves.