Ancient solutions of the affine normal flow

John Loftin Rutgers University Mao-Pei Tsui The University of Toledo

Differential Geometry mathscidoc:1609.10049

Journal of Differential Geometry, 78, (1), 113-162, 2008
We construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the existence of a hyperbolic affine sphere asymptotic to the boundary of a convex cone containing no lines, which is originally due to Cheng-Yau. The main techniques are local second-derivative estimates for a parabolic Monge-Amp`ere equation modeled on those of Ben Andrews and Guti´errez-Huang, a decay estimate for the cubic form under the affine normal flow due to Ben Andrews, and a hypersurface barrier due to Calabi.
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@inproceedings{john2008ancient,
  title={ANCIENT SOLUTIONS OF THE AFFINE NORMAL FLOW},
  author={John Loftin, and Mao-Pei Tsui},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909104057118375706},
  booktitle={Journal of Differential Geometry},
  volume={78},
  number={1},
  pages={113-162},
  year={2008},
}
John Loftin, and Mao-Pei Tsui. ANCIENT SOLUTIONS OF THE AFFINE NORMAL FLOW. 2008. Vol. 78. In Journal of Differential Geometry. pp.113-162. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909104057118375706.
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