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The topological uniqueness of complete minimal surfaces of finite topological type

William H Meeks III Shing-Tung Yau

Differential Geometry mathscidoc:1912.43597

Topology, 31, (2), 305-316, 1992.4
IN 1970 Lawson [22] proved that two embedded closed diffeomorphic minimal surfaces in the unit three-dimensional sphere S3 in lRJ are ambiently isotopic in S3. Lawson proved this theorem by first proving that an embedded orientable closed minimal surface of genus y in a closed orientable Riemannian three-manifold Mwith positive Ricci curvature disconnects M-into two genus-8 handlebodics. A result of Frankel [7] was used to prove this. Lawson then applied a deep result of Waldhauscn [33] that states that decompositions of SJ into two genus-cl handlebodies arc unique up to ambient isotopy. More prcciscly. Waldhauscns uniqucncss thcorcm states that whenever a closed surface of genus y in S separates SJ into handlcbodics, then the embedding of the surface is as simple as possible; in other words, the surface is obtained from a two-sphere Sz c Sby adding handles in an unknotted manner.
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@inproceedings{william1992the,
  title={The topological uniqueness of complete minimal surfaces of finite topological type},
  author={William H Meeks III, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204327521854161},
  booktitle={Topology},
  volume={31},
  number={2},
  pages={305-316},
  year={1992},
}
William H Meeks III, and Shing-Tung Yau. The topological uniqueness of complete minimal surfaces of finite topological type. 1992. Vol. 31. In Topology. pp.305-316. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204327521854161.
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