Let $(M,g)$ be a compact Riemannian manifold (e.g. a Calabi–Yau $m$-fold) and $\{L^{t}:t\in[0,T)\}$ a family of compact immersed submanifolds in $M$ (e.g. Lagrangians) satisfying mean curvature flow. We say that the family has a finite time singularity at $t=T$ if the flow cannot be smoothly continued to $[0,T+\epsilon)$ for any $\epsilon>0$. As in Wang [71, Lem. 5.1] this implies that $\mathop{\rm lim\,sup}_{t\rightarrow T}\|A^{t}\|_{C^{0}}\rightarrow\infty$, where $A^{t}$ is the second fundamental form of $L^{t}$.

We call such a finite time singularity of type I if $\|A^{t}\|_{C^{0}}^{2}\leqslant\penalty C/(T-t)$ for some $C>0$ and all $t\in[0,T)$. Otherwise we call the singularity of type II.

We call $x\in M$ a singular point of the flow if $\mathop{\rm lim\,sup}_{t\rightarrow T}\|A^{t}|_{U\cap L^{t}}\|_{C^{0}}=\infty$ for all open neighbourhoods $U$ of $x$ in $M$.