NONHOMOGENEOUS GAUSS CURVATURE FLOWS

We study the expansion of a smooth closed convex hypersurface in euclidean space by a nonhomogeneous function of the Gaussian curvature. The contraction and bi-directional cases are also treated briefly. Given a closed convex n-dimensional hypersurface M0 in R, we shall consider its expansion along its outward normal vector direction with speed equal to a given function F (1/K), where K is the Gaussian curvature of the convex hypersurface and F : R+ → R+ is a positive smooth increasing function, i.e., F ′ > 0 everywhere. Using the support function u(x, t) of the convex hypersurface, we can get the following Monge-Ampére equation on S ∂u ∂t = F (det(∇i∇ju+ ugij)) on S n × [0, T ), where gij is the standard metric on S n and ∇ is the covariant differentiation. Under the concavity assumption on the speed F , we show that the initial hypersurface M0 will remain smooth, convex, and expand to infinity with its shape becoming round asymptotically, which generalizes the results of John Urbas where he considered the homogeneous case F (z) = z.