Limits of Solutions to a Parabolic Monge-Ampere Equation

TL;DR

This paper investigates the behavior of solutions to a parabolic Monge-Ampere equation on the sphere, establishing long-term existence, smoothing effects, and classifying ancient solutions as ellipsoids or paraboloids.

Contribution

It provides new PDE-based analysis of the limits of solutions to a Monge-Ampere equation with unbounded initial data, extending previous affine flow results.

Findings

Proves long-time existence and smoothing for general initial data.

Characterizes ancient solutions as ellipsoids or paraboloids.

Uses estimates from Andrews, Gutierrez-Huang, and Calabi barriers.

Abstract

We present the results from our earlier paper (arXiv:math/0602484) on the affine normal flow on noncompact convex hypersurfaces in affine space from a more PDE point of view, emphasizing the estimates involved. Our results concern the limits of solutions to a parabolic Monge-Ampere equation on $S^n$, where a sequence of smooth strictly convex initial value functions increase monotonically to a limiting initial value function which is infinite on at least a hemisphere of $S^n$. We prove long-time existence and instantaneous smoothing for quite general initial data, and we characterize ancient solutions as ellipsoids or paraboloids. We make essential use of estimates of Andrews and Gutierrez-Huang, and barriers due to Calabi.