Affine Normal Flow of Noncompact Hypersurfaces


John Loftin

14:00:00 - 15:00:00

浦大邦講堂 , Institute of Atomic and Molecular Sciences, Dr. Poe Lecture Hall

The affine normal flow is a parabolic flow of convex hypersurfaces in Rn+1 which is invariant under volume-preserving affine transfomations. In joint work with M.P. Tsui, 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-Amp\\`ere 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.