Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-04-22 , DOI: 10.1016/j.jfa.2021.109060 Heiko Kröner
Generalizing results of Chou and Wang [12] we study the flows of the leaves of a foliation of consisting of uniformly convex hypersurfaces in the direction of their outer normals with speeds . For quite general functions F of the principal curvatures of the flow hypersurfaces and f a smooth and positive function on (considered as a function of the normal) we show that there is a distinct leaf in this foliation with the property that the flow starting from converges to a translating solution of the flow equation. When starting the flow from a leave inside it shrinks to a point and when starting the flow from a leave outside 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 . 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.
中文翻译:
由一般曲率函数的对数给出的以正常速度使叶的叶子流动
概括Chou和Wang [12]的结果,我们研究了叶子的流动 叶的 由均匀的凸超曲面在其外法线方向上以速度组成 。对于一般函数,流动超曲面的主曲率F和f上的光滑正函数 (作为法线的函数),我们表明有一个明显的叶子 在这种叶状结构中,流动从 收敛到流量方程的平移解。从内部离开开始流动时 它缩小到一个点,当从外面的休假开始流动时 它扩展到无穷大。尽管[12]认为这种机制的F等于高斯曲率,但我们允许F成为基本对称多项式等。此外,我们表明,这种行为至少在旋转对称且同质度一个曲率函数的情况下相对于放宽某些假设是鲁棒的。