Ancient solutions of the affine normal flow (Q2427047)

The authors construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solutions) or paraboloids (translating solutions). They also provide a new proof for 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 Gutierrez-Huang, a decay estimate for the cubic form under the affine normal flow due to Ben Anderws, and a hypersurface barrier due to Calabi.