A logarithmic Gauss curvature flow and the Minkowski problem

Abstract

Let X0 be a smooth uniformly convex hypersurface and f a postive smooth function in Sn. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX0 along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ∗>0 such that if θ<θ∗, they shrink to a point in finite time and, if θ>θ∗, they expand to an asymptotic sphere. Finally, when θ=θ∗, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).