A class of generalized fully nonlinear curvature flows and its applications

TL;DR

This paper introduces a generalized class of fully nonlinear curvature flows involving elementary symmetric functions of principal curvatures, establishing long-term existence and convergence results in Euclidean space.

Contribution

It extends curvature flow theory to a broader class involving the $k$-th elementary symmetric function, providing new convergence results without constraints for the case $k=n-1$.

Findings

Proved long-time existence of the flow for $1 \,\leq\, k \leq n-1$.

Established convergence of the flow under certain initial conditions.

Achieved convergence results for the case $k=n-1$ without additional constraints.

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 $1\leq k\leq n-1$. For $1\leq k< n-1$, 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.