Journal of Differential Geometry

Ancient solutions of the affine normal flow

John Loftin and Mao-Pei Tsui

Full-text: Open access

Abstract

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.

Article information

Source
J. Differential Geom., Volume 78, Number 1 (2008), 113-162.

Dates
First available in Project Euclid: 10 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1197320604

Digital Object Identifier
doi:10.4310/jdg/1197320604

Mathematical Reviews number (MathSciNet)
MR2406266

Zentralblatt MATH identifier
1146.53038

Citation

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. https://projecteuclid.org/euclid.jdg/1197320604


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