Limits of Solutions to a Parabolic Monge-Ampère Equation

In [10], we study solutions to the affine normal flow for an initial hypersurface L ⊂ R which is a convex, properly embedded, noncompact hypersurface. The method we used was to consider an exhausting sequence Li of smooth, strictly convex, compact hypersurfaces so that each Li is contained in the convex hull of Li+1 for each i, and so that Li → L locally uniformly. If the compact Li is the initial hypersurface, the affine normal flow Li(t) is well-defined for all time t from 0 to the extinction time Ti [7]. Then for all positive t, we define the affine normal flow for initial hypersurface L as a limit L(t) = limi→∞Li(t). Ben Andrews extensively studies the affine normal flow for compact initial hypersurfaces [1, 2]. The method of proof in [10] is to consider the support functions sLi = si and to take the limit as i → ∞. For each Y ∈ R, the support function is defined by s(Y ) = sL(Y ) = sup x∈L 〈x, Y 〉,