  Given a family of smooth immersions $F_t: M^n\to N^{n+1}$ of closed
hypersurfaces in a locally symmetric Riemannian manifold $N^{n+1}$ with bounded
geometry, moving by the mean curvature flow, we show that at the first finite
singular time of the mean curvature flow, certain subcritical quantities
concerning the second fundamental form blow up. This result not only
generalizes a recent result of Le-Sesum and Xu-Ye-Zhao, but also extends the
latest work of N. Le in the Euclidean case (arXiv: math.DG/1002.4669v2).
