2006/02/22 by John Loftin, Loftin, John, Mao-Pei Tsui +1 · 2 citations
Mathematics · #53A15 #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:53A15 #msc:53C44
paper · pdf · doi:10.48550/arxiv.math/0602484
A corrollary retracted, and a remark and some typos fixed
arxiv created 2006/07/31 · arxiv updated 2009/12/01
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.