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The space of embedded minimal surfaces of fixed genus in a 3-manifold II; Multi-valued graphs in disks

Tobias H. Colding·William P. Minicozzi II
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Abstract

This paper is the second in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in $\RR^3$. We show here that if the curvature of such a disk becomes large at some point, then it contains an almost flat multi-valued graph nearby that continues almost all the way to the boundary.

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Tobias H. Colding, William P. Minicozzi II. 2002-10-07. The space of embedded minimal surfaces of fixed genus in a 3-manifold II; Multi-valued graphs in disks. https://arxiv.org/abs/math/0210086

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