A logarithmic Gauss curvature flow and the Minkowski problem

Abstract

Let $X_0$ be a smooth uniformly convex hypersurface and $f$ a postive smooth function in $S^n$. We study the motion of convex hypersurfaces $X(\cdot ,t)$ with initial $X(\cdot ,0)=\theta X_0$ along its inner normal at a rate equal to $\log (K/f)$ where $K$ is the Gauss curvature of $X(\cdot ,t)$. We show that the hypersurfaces remain smooth and uniformly convex, and there exists ${\theta }^{\ast }>0$ such that if $\theta <{\theta }^{\ast }$, they shrink to a point in finite time and, if $\theta >{\theta }^{\ast }$, they expand to an asymptotic sphere. Finally, when $\theta ={\theta }^{\ast }$, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on $f(x)$.