Flowing the leaves of a foliation with normal speed given by the logarithm of general curvature functions

Abstract

Generalizing results of Chou and Wang \cite{1} we study the flows of the leaves $(M_{\Theta})_{\Theta>0}$ of a foliation of $\mathbb{R}^{n+1}\setminus \{0\}$ consisting of uniformly convex hypersurfaces in the direction of their outer normals with speeds $-\log(F/f)$. For quite general functions $F$ of the principal curvatures of the flow hypersurfaces and $f$ a smooth and positive function on $S^n$ (considered as a function of the normal) we show that there is a distinct leaf $M_{\Theta_{*}}$ in this foliation with the property that the flow starting from $M_{\Theta_{*}}$ converges to a translating solution of the flow equation. Furthermore, when starting the flow from a leave inside $M_{\Theta_{*}}$ it shrinks to a point and when starting the flow from a leave outside $M_{\Theta_{*}}$ it expands to infinity. While \cite{1} considered this mechanism with $F$ equal to the Gauss curvature we allow $F$ to be among others the elementary symmetric polynomials $H_k$. We, furthermore, show that such kind of behavior is robust with respect to relaxing certain assumptions at least in the rotationally symmetric and homogeneous degree one curvature function case.