(a) The ‘neck pinching’ behaviour of Conjecture 3.16 is inverse to the ‘opening a neck’ behaviour of §3.4. So, for example, we can imagine a flow $\{L^{t}:t\in[0,\infty)\}$ satisfying the programme of §3.2, with two singular times $0<T_{1}<T_{2}$, which starts with a single $L^{t}$ for $0\leqslant\penalty t<T_{1}$, undergoes a ‘neck pinch’ at $t=T_{1}$ and becomes a union $L^{t}=L^{t}_{1}\cup L^{t}_{2}$ of Lagrangians $L^{t}_{1},L^{t}_{2}$ intersecting at one point $p^{t}$ for $T_{1}<t<T_{2}$, and then at $t=T_{2}$ ‘opens the neck’ at $p^{t}$ and turns back into a single Lagrangian $L^{t}$ for $t>T_{2}$.

Note that these inverse singular behaviours involve different (though related) geometric local models, Lawlor necks $L_{\boldsymbol{\phi},A}$ and Joyce–Lee–Tsui expanders $L_{\boldsymbol{\phi}}^{\alpha}$. We do not just naïvely run the local picture for the flow in reverse. Note too that ‘neck pinching’ works only for $m\geqslant 2$, whereas ‘opening necks’ works for $m\geqslant 1$, so when $m=1$, ‘opening necks’ has no inverse behaviour.

In a similar way, the author expects that many types of finite time singularity possible in the programme of §3.2 should have a corresponding inverse type, so that changes in the topology of $L^{t}$, and other qualitative features, are reversible. An exception to this is that when $m=1$, the flow can only decrease the number of self-intersection points, making the curve ‘less immersed’.

Also, when $m=2$ the local systems $E^{t^{\prime}}$ for $t^{\prime}\in(T-\epsilon,T)$ could have nontrivial holonomy around the ‘neck’. If so, the definition of $E^{t}$ for $t\in[T,T+\epsilon)$ in part (vi) no longer makes sense, since we cannot extend $E^{t^{\prime}}$ over $p^{t}$ in $L^{t}_{\pm}$.

One conclusion is that for $m=2$, though neck pinches should be generic under Hamiltonian perturbations, they may be nongeneric (and of index 1) under Lagrangian perturbations, since Lagrangian perturbations may allow the flow to wander in ${\mathbin{\cal F}}={\mathbin{\mathbb{R}}}^{2}\setminus\{0\}$ rather than ${\mathbin{\cal F}}_{\rm exact}={\mathbin{\mathbb{R}}}\setminus\{0\}$, and will only hit the singularity $0\in{\mathbin{\mathbb{R}}}^{2}$ in real codimension 1 amongst initial Lagrangians.

We can also ask: if Lagrangian MCF $\{L^{t}:t\in(T-\epsilon,T)\}$ develops a singularity as $t\rightarrow T$ modelled on Lawlor necks $L_{\boldsymbol{\phi},A}$ for $A\in(0,\infty)\subset{\mathbin{\cal F}}_{\rm exact}={\mathbin{\mathbb{R}}}\setminus\{0\}$, rather than continuing for $t>T$ using immersed SL 2-folds as in Conjecture 3.16, why not continue using Lawlor necks $\tilde{L}_{\boldsymbol{\phi},A}$ for $A\in(-\infty,0)\subset{\mathbin{\cal F}}_{\rm exact}={\mathbin{\mathbb{R}}}\setminus\{0\}$, immediately opening the neck again, in a similar way to §3.4?

The author expects that this is the correct thing to do if $E^{t^{\prime}}$ for $t^{\prime}\in(T-\epsilon,T)$ has nontrivial holonomy around the ‘neck’. But in the trivial holonomy case, it would change the isomorphism class of $(L^{t},E^{t},b^{t})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$, and so should be avoided according to the philosophy of §3.2.