Open Access
January 2008 Ancient solutions of the affine normal flow
John Loftin, Mao-Pei Tsui
J. Differential Geom. 78(1): 113-162 (January 2008). DOI: 10.4310/jdg/1197320604


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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John Loftin. Mao-Pei Tsui. "Ancient solutions of the affine normal flow." J. Differential Geom. 78 (1) 113 - 162, January 2008.


Published: January 2008
First available in Project Euclid: 10 December 2007

zbMATH: 1146.53038
MathSciNet: MR2406266
Digital Object Identifier: 10.4310/jdg/1197320604

Rights: Copyright © 2008 Lehigh University

Vol.78 • No. 1 • January 2008
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