Normal curvature bounds along the 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.