Let $(M,J,g,\Omega)$ be a compact Calabi–Yau $m$-fold and $\{L^{t}:t\in[0,T)\}$ a family of compact, immersed, graded Lagrangians in $M$ satisfying Lagrangian MCF, with a finite time singularity at $t=T,$ and a singular point at $x\in M$. Here are broad descriptions of two classes of such singularities:

• (a)

  Let $U$ be a small open neighbourhood of $x$ in $M,$ which we identify with a small open neighbourhood of $0$ in ${\mathbin{\mathbb{C}}}^{m}=T_{x}M,$ and $\epsilon>0$ be small. Then $L^{t}\cap U$ approximates a closed, exact SL $m$-fold in ${\mathbin{\mathbb{C}}}^{m}$ for $t\in(T-\epsilon,T)$.

  Since SL $m$-folds are stationary points of LMCF, to ‘first order’ $L^{t}\cap U$ is constant in $t,$ but to ‘second order’ $L^{t}\cap U$ wanders slowly in the moduli space of closed, exact SL $m$-folds in ${\mathbin{\mathbb{C}}}^{m},$ until at time $t=T$ it hits a singular SL $m$-fold. This ‘wandering’ is driven by ‘outside influences’ from the whole of $L^{t},$ not just from $L^{t}\cap U$.

  For example, if $N$ is an exact asymptotically conical SL $m$-fold in ${\mathbin{\mathbb{C}}}^{m},$ we could have $L^{t}\cap U\approx f(t)\cdot N$ for $t\in(T-\epsilon,T),$ where $f:(T-\epsilon,T)\rightarrow(0,\infty)$ is smooth with $f(t)\rightarrow 0$ as $t\rightarrow T$.

• (b)

  Let $U,\epsilon$ be as in (a). Then $L^{t}\cap U$ approximates a closed, exact LMCF translator in ${\mathbin{\mathbb{C}}}^{m}=T_{x}M$ for $t\in(T-\epsilon,T)$. To ‘first order’ $L^{t}\cap U$ moves by translation in ${\mathbin{\mathbb{C}}}^{m}=T_{x}M,$ since it approximates a translating soliton. But to second order it also wanders slowly in the moduli space of closed, exact LMCF translators in ${\mathbin{\mathbb{C}}}^{m},$ driven by ‘outside influences’ from the whole of $L^{t},$ until at time $t=T$ it hits a singular soliton.

  For example, if $N$ is an exact LMCF translator in ${\mathbin{\mathbb{C}}}^{m}$ with translating vector $v\in{\mathbin{\mathbb{C}}}^{m},$ we could have $L^{t}\cap U\approx f(t)\cdot N+g(t)\cdot v$ for $t\in(T-\epsilon,T),$ where $f,g:(T-\epsilon,T)\rightarrow(0,\infty)$ are smooth with $f(t)\rightarrow 0$ as $t\rightarrow T$.