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 $|{\bar{\nabla }}^k\overline{Rm}|\le \bar{C}$ for $k\le 2$, and suppose there is an isometric immersion $F_0:M^n\to {\bar{M}}^{n+1}$ with bounded second fundamental form. Let $F_t:M^n\to {\bar{M}}^{n+1}$ ($t\in [0,T]$) 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.

arXiv comment4 pages, some corrections