Limits of Solutions to a Parabolic Monge-Ampere Equation

2008/02/01 by John Loftin, Loftin, John, Mao‐Pei Tsui +2
Mathematics · #35K55 #53A15 #53C44 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.AP #msc:35K55 #msc:53A15 #msc:53C44

paper · pdf · doi:10.48550/arxiv.0802.0208

arxiv created 2008/02/01 · openalex publication_date 2008/02/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Sⁿ, 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 Sⁿ. 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.

Citations

• Ancient Solutions of the Affine Normal Flow

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