Abstract
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 \itnonsimply 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.
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. 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.
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