Constrained willmore tori in the 4–sphere

Christoph Bohle Technische Universit¨at Berlin

Differential Geometry mathscidoc:1609.10193

Journal of Differential Geometry, 86, (1), 71-131, 2010
We prove that a constrained Willmore immersion of a 2–torus into the conformal 4–sphere S4 is of “finite type”, that is, has a spectral curve of finite genus, or of “holomorphic type” which means that it is super conformal or Euclidean minimal with planar ends in R4 = S4\{1} for some point 1 2 S4 at infinity. This implies that all constrained Willmore tori in S4 can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of Hitchin’s approach [19] to the study of harmonic tori in S3.
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@inproceedings{christoph2010constrained,
  title={CONSTRAINED WILLMORE TORI IN THE 4–SPHERE},
  author={Christoph Bohle},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911224009259834850},
  booktitle={Journal of Differential Geometry},
  volume={86},
  number={1},
  pages={71-131},
  year={2010},
}
Christoph Bohle. CONSTRAINED WILLMORE TORI IN THE 4–SPHERE. 2010. Vol. 86. In Journal of Differential Geometry. pp.71-131. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911224009259834850.
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