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 $M_{{\Theta }_{\ast }}$ in this foliation with the property that the flow starting from $M_{{\Theta }_{\ast }}$ converges to a translating solution of the flow equation. Furthermore, when starting the flow from a leave inside $M_{{\Theta }_{\ast }}$ it shrinks to a point and when starting the flow from a leave outside $M_{{\Theta }_{\ast }}$ 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.

34 pages

References in corpus (2)

• Complete translating solitons to the mean curvature flow in ${\mathbb{R}}^3$ with nonnegative mean curvature
• Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature