Entropy and a convergence theorem for Gauss curvature flow in high dimension

Abstract

We prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in $C^{\infty }$-topology to a smooth strictly convex soliton as $t$ goes to infinity is obtained as a consequence of these estimates together with an earlier result of Andrews. The estimates are established via the study of an entropy functional for convex bodies.