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The space of embedded minimal surfaces of fixed genus in a 3-manifold. III: Planar domains. (English) Zbl 1076.53068
The paper under review is the third one in a series where the authors describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed $$3$$-manifold [see Part I, Ann. Math. (2) 160, No. 1, 27–68 (2004; Zbl 1070.53031); II, ibid. 160, No. 1, 69–92 (2004; Zbl 1070.53032); IV, ibid. 160, No. 2, 573–615 (2004; Zbl 1076.53069)].
There are two main themes in this paper. The first is that stability leads to improved curvature estimates. This allows them to find large graphical regions. These graphical regions lead to two possiblities: (1) they “close up” to form a graph, (2) a multi-valued graph froms. The second theme is that in certain important cases they can rule out the formation of multi-valued graph.

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 58D10 Spaces of embeddings and immersions 58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
##### Keywords:
embedded minimal surface; genus; planar domain
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