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The space of embedded minimal surfaces of fixed genus in a 3-manifold. II: Multi-valued graphs in disks. (English) Zbl 1070.53032

This important paper is the second in a series where authors give a description of the space of all embedded minimal surfaces of fixed genus in a closed Riemannian 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 \(\mathbb{R}^3\). The authors prove 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. This is proven by showing the existence of small multi-valued graphs near points of large curvature and then using the extension result for multi-valued graphs, proven in the first paper in this series (see [Ann. Math. (2) 160, No.1, 27–68 (2004; Zbl 1070.53031)]).

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58D10 Spaces of embeddings and immersions
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)

Citations:

Zbl 1070.53031
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