Chapter IX Group Actions on R3

William H Meeks III Shing-Tung Yau

Differential Geometry mathscidoc:1912.43729

112, 167-179, 1984.1
The chapter discusses the group actions on R<sup>3</sup>. The Smith conjecture has many equivalent forms, and each of these forms has various consequences and generalizations. The Smith conjecture is a structure theorem about symmetries of the product of a compact surface with an interval. Here the interval may be closed or open. The usual Smith conjecture is equivalent to proving the smooth <i>Z<sub>n</sub></i> actions on <i>S</i><sup>2</sup> [0, 1] are conjugate to actions that preserve the product structure. This generalized Smith conjecture represents the belief that all the symmetries of the product of a compact surface with an interval actually arise from the symmetries of the surface extended trivially to the product structure. The chapter presents how to apply minimal surfaces to study the generalized Smith conjecture on <i>S</i><sup>2</sup> <i>I</i>, where <i>I</i> is an open or closed interval. A geodesic version of the loop theorem and a new equivariant
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@inproceedings{william1984chapter,
  title={Chapter IX Group Actions on R3},
  author={William H Meeks III, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205320443200293},
  volume={112},
  pages={167-179},
  year={1984},
}
William H Meeks III, and Shing-Tung Yau. Chapter IX Group Actions on R3. 1984. Vol. 112. pp.167-179. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205320443200293.
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