Journal of Differential Geometry

Ancient solutions of the affine normal flow

John Loftin and Mao-Pei Tsui

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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-Ampère equation modeled on those of Ben Andrews and Gutiérrez-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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J. Differential Geom., Volume 78, Number 1 (2008), 113-162.

First available in Project Euclid: 10 December 2007

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Loftin, John; Tsui, Mao-Pei. Ancient solutions of the affine normal flow. J. Differential Geom. 78 (2008), no. 1, 113--162. doi:10.4310/jdg/1197320604.

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