Some extensions of the mean curvature flow in Riemannian manifolds

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).