Higher order geometric flow of hypersurfaces in a Riemannian manifold

TL;DR

This paper studies high order geometric flows of hypersurfaces in Riemannian manifolds, extending previous Euclidean results and establishing long-time existence under curvature and injectivity radius conditions.

Contribution

It generalizes Mantegazza's Euclidean hypersurface flow results to curved ambient spaces with specific geometric bounds.

Findings

Flow exists for all time under curvature bounds

Extension of Euclidean results to Riemannian manifolds

Conditions on injectivity radius and sectional curvature ensure long-term flow stability

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 $\infty$.