A class of generalized fully nonlinear curvature flows and its applications

Abstract

In this paper, we concern a generalized fully nonlinear curvature flow involving $k$-th elementary symmetric function for principal curvature radii in Eulidean space $\rnnn$, $k$ is an integer and $1\le k\le n-1$. For $1\le k<n-1$, based on some initial data and constrains on smooth positive function defined on the unit sphere $\sn$, we obtain the long time existence and convergence of the flow. Especially, the same result shall be derived for $k=n-1$ without any constraint on the smooth positive function.