Normal curvature bounds along the mean curvature flow

TL;DR

This paper establishes bounds on the variation of normal curvature along the mean curvature flow of submanifolds in Riemannian manifolds with bounded curvature derivatives.

Contribution

It extends curvature bound results to mean curvature flow, showing normal curvature bounds evolve at a controlled rate in bounded curvature ambient spaces.

Findings

Normal curvature bounds are maintained along the flow.

Supremum and infimum of normal curvature vary at a bounded rate.

Analogue of Ricci flow curvature bounds applied to mean curvature flow.

Abstract

Let $(M^n,g_0)$ and $(\bar{M}^{n+1},\bar{g})$ be complete Riemannian manifolds with $|\bar{\nabla}^k\bar{Rm}|\le \bar{C}$ for $k \le 2$, and suppose there is an isometric immersion $F_0: M^n \rightarrow \bar{M}^{n+1}$ with bounded second fundamental form. Let $F_t: M^n \rightarrow \bar{M}^{n+1}$ ($t\in [0,T]$) be a family of immersions evolving by mean curvature flow with initial data $F_0$ and with uniformly bounded second fundamental forms. We show that the supremum and infimum of the normal curvature of the immersions $F_t$ vary at a bounded rate. This is an analogue of a result of Rong and Kapovitch on Ricci flow.