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Chapter VIII The Equivariant Loop Theorem for Three-Dimensional Manifolds and a Review of the Existence Theorems for Minimal Surfaces

Shing-Tung Yau William H Meeks III

Differential Geometry mathscidoc:1912.43609

112, 153-163, 1984.1
The chapter discusses the equivariant loop theorem for three-dimensional manifolds that is needed in settling the Smith conjecture and reviews the existence theorems for minimal surfaces. The equivariant version of the loop theorem says that there are a finite number of properly embedded disks in <i>M</i> that satisfy the required properties. The loop theorem respects the action of the group <i>G</i> in a suitable manner. The chapter puts a metric on <i>M</i> so that the group <i>G</i> acts isometrically and so that <i>M</i> is convex with respect to the outward normal. Then with respect to this metric, the existence of an immersed disk <i>D</i><sub>1</sub> is demonstrated, in <i>M</i> whose boundary <i> D<sub>1</sub></i>, represents a nontrivial element in <sub>1</sub>(<i>S</i>) and whose area is minimal among all such disks. The chapter describes Morrey's solution for the plateau problem in a general Riemannian manifold. The existence theorem for manifolds with boundary
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@inproceedings{shing-tung1984chapter,
  title={Chapter VIII The Equivariant Loop Theorem for Three-Dimensional Manifolds and a Review of the Existence Theorems for Minimal Surfaces},
  author={Shing-Tung Yau, and William H Meeks III},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204414367006173},
  volume={112},
  pages={153-163},
  year={1984},
}
Shing-Tung Yau, and William H Meeks III. Chapter VIII The Equivariant Loop Theorem for Three-Dimensional Manifolds and a Review of the Existence Theorems for Minimal Surfaces. 1984. Vol. 112. pp.153-163. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204414367006173.
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