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

Abstract

Generalizing results of Chou and Wang [12] we study the flows of the leaves (MΘ)Θ>0 of a foliation of Rn+1∖{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 Sn (considered as a function of the normal) we show that there is a distinct leaf MΘ⁎ in this foliation with the property that the flow starting from MΘ⁎ converges to a translating solution of the flow equation. When starting the flow from a leave inside MΘ⁎ it shrinks to a point and when starting the flow from a leave outside MΘ⁎ it expands to infinity. While [12] considered this mechanism with F equal to the Gauss curvature we allow F to be among others the elementary symmetric polynomials Hk. Furthermore, we 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.