Geodesic
The space of embedded minimal surfaces of fixed genus in a 3-manifold V; Fixed genus
Tobias H. Colding1 ; William P. Minicozzi II2
1 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY
2 Department of Mathematics, Johns Hopkins University, Baltimore, MD
Annals of mathematics, Tome 181 (2015) no. 1, pp. 1-153.

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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.
DOI : 10.4007/annals.2015.181.1.1

Tobias H. Colding 1 ; William P. Minicozzi II 2

1 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY
2 Department of Mathematics, Johns Hopkins University, Baltimore, MD 
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Tobias H. Colding; William P. Minicozzi II. The space of embedded minimal surfaces of fixed genus in a 3-manifold V; Fixed genus. Annals of mathematics, Tome 181 (2015) no. 1, pp. 1-153. doi : 10.4007/annals.2015.181.1.1. http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.181.1.1/

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