Ancient Solutions of the Affine Normal Flow

Details Ancient Solutions of the Affine Normal Flow Ancient Solutions of the Affine Normal Flow

2006-02-22

arxiv.org/abs/math/0602484v2

Mathematics

Differential Geometry

A corrollary retracted, and a remark and some typos fixed

Scientific paper

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-Ampere equation modeled on those of Ben Andrews and Gutierrez-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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