In this work, we consider the mean curvature flow of compact submanifolds of Riemannian manifolds. If the flow becomes singular in nite time, we show how to produce a smooth singularity model, the "smooth blow-up" of the singularity. This construction relies upon a compactness theorem for families of submanifolds with bounded second fundamental form which we establish.Using the smooth blow-up, we establish that if the contraction of the mean curvature with the second fundamental form is bounded, the flow may be continued. We also show that in case the singularity is of type I, the mean curvature must blow up and that in the type II case, the mean curvature must blow up, if at all, at a strictly slower rate than the full second fundamental form.We also use the smooth blow-up to investigate Lagrangian mean curvature flow in Calabi-Yau manifolds. In particular we show that the singularities of the Lagrangian mean curvature flow are modelled either by zero Maslov class or monotone Lagrangian flows in Euclidean space.
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