Higher order geometric flow of hypersurfaces in a Riemannian manifold

Abstract

In this paper, we consider the high order geometric flows of a submanifolds $M$ in a complete Riemannian manifold $N$ with $\dim(N)=\dim(M)+1=n+1$, which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on $N$. Precisely, we show that if $m\in\mathbb{N}$ is strictly larger than the integer part of $n/2$ and $\varphi(t)$ is a immersion for all $t\in[0,T)$ and if $\mathfrak{F}_m(\varphi_0)$ is bounded by a constant which relies on the injectivity radius $\bar{R}>0$ and sectional curvature $\bar{K}_{\pi}(\bar{K}_{\pi}\leqslant1)$ of $N$ , then $T$ must be $\infty$.