Title: Nonhomogeneous Gauss curvature flows

Authors: Bennett Chow and Dong-Ho Tsai

Issue: Volume 47 (1998), Issue 3, 965-994

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.\par Given a closed convex $n$-dimensional hypersurface $M_0$ in $\mathbb{R}^{n+1}$, 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 $F : \mathbb{R}_+ \to \mathbb{R}_+$ 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-Amp\`ere equation on $S^n$: \[ \frac{\partial u}{\partial t} = F(\mathrm{det}(\nabla_i \nabla_j u + ug_{ij}))\quad\text{on } S^n \times [0,T) \] where $g_{ij}$ is the standard metric on $S^n$ and $\nabla$ is the covariant differentiation.\par Under the concavity assumption on the speed $F$, we show that the initial hypersurface $M_0$ 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 $F(z) = z^{1/n}$.