A generalization of a theorem by Calabi to the parabolic Monge-Ampere equation

Abstract

A celebrated result of Calabi [C], that generalizes a two dimensional theorem by Jorgens [J], asserts that if u is a C5 convex solution of the elliptic MongeAmpere equation detD2u = 1 in Rn and n ≤ 5 then u is a quadratic polynomial. This statement was extended to all dimensions by Pogorelov [P2] and Cheng and Yau [Ch-Y]. Recently, Caffarelli [Ca3] proved, by using the regularity theory for the Monge-Ampere equation developed in the fundamental papers [Ca1] and [Ca2], that this result holds true for viscosity solutions. The purpose of this article is to investigate the validity of results of the same nature for solutions of the parabolic Monge-Ampere equation −ut detD2u = 1 in Rn × (−∞, 0]. This type of differential operator was first considered by Krylov [K2]. It also appears in connection with the problem of the deformation of a surface by means of its Gauss-Kronecker curvature. Indeed, Tso [T] solved this problem by noting that the support function to the surface that is deforming satisfies an initial value problem involving that parabolic operator. The function u : Rn × (−∞, 0]→ R, u = u(x, t), is called parabolically convex if it is continuous, convex in x and nonincreasing in t. By D2u(x, t) we denote the matrix of second derivatives of u with respect to x and Du denotes the gradient of u with respect to x. We use the standard notation C2k,k(Ω) to denote the class of functions u such that the derivatives Di xD j tu are continuous in Ω for i+ 2j ≤ 2k. We set R − = R n × (−∞, 0]. The main result of this paper is the following theorem.