Abstract
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 M <sub>0</sub> in ℝ <sup>n+1</sup> , 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 <sub>+</sub> → ℝ <sub>+</sub> 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 <sub>n</sub> ∂u/∂t = F (det(∇ <sub>i</sub> ∇ <sub>j</sub> u+ug <sub>ij</sub> )) on S <sub>n</sub> × [0, T), where g <sub>ij</sub> is the standard metric on S <sup>n</sup> and ∇ is the covariant differentiation. Under the concavity assumption on the speed F, we show that the initial hypersurface M <sub>0</sub> 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 <sup>1/n</sup> .