The following behaviour, which we call a ‘neck pinch’, can occur in Lagrangian MCF with surgeries in Calabi–Yau $m$-folds for $m\geqslant 2,$ as in §3.2. Furthermore, ‘neck pinches’ are a generic singularity. That is, if Lagrangian MCF beginning from $L^{0}$ develops a neck pinch, then Lagrangian MCF beginning from any sufficiently small Hamiltonian perturbation $\tilde{L}^{0}$ of $L^{0}$ also develops a neck pinch.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold, and extend $D^{b}{\mathbin{\mathscr{F}}}(M)$ to include immersed Lagrangians, as in [2]. Suppose $\{(L^{t},E^{t}):t\in(T-\epsilon,T+\epsilon)\}$ for $\epsilon>0$ small is a family of immersed Lagrangian branes in $M$ with $HF^{*}$ unobstructed, and $\{b^{t}:t\in(T-\epsilon,T+\epsilon)\}$ a corresponding family of bounding cochains, satisfying the following conditions:

• (i)

  The $(L^{t},E^{t},b^{t})$ for $t\in(T-\epsilon,T+\epsilon)$ are all isomorphic in $D^{b}{\mathbin{\mathscr{F}}}(M)$.

• (ii)

  When $t<T,$ $L^{t},E^{t}$ depend smoothly on $t\in(T-\epsilon,T),$ and $\{(L^{t},E^{t}):t\in(T-\epsilon,T)\}$ satisfies Lagrangian MCF, with a finite time singularity at $t=T,$ with one singular point $p\in M$.

  Similarly, when $t\geqslant T,$ $L^{t},E^{t}$ depend smoothly on $t\in[T,T+\epsilon),$ and $\{(L^{t},E^{t}):t\in[T,T+\epsilon)\}$ satisfies Lagrangian MCF. The topology of $L^{t}$ for $t\in(T-\epsilon,T+\epsilon)$ changes discontinuously at $t=T$. Nonetheless, the family $\{L^{t}:t\in(T-\epsilon,T+\epsilon)\}$ is continuous at $t=T$ in a suitable sense, e.g. as graded Lagrangian integral currents in Geometric Measure Theory.

• (iii)

  Identifying $M$ near $p$ with $T_{p}M\cong{\mathbin{\mathbb{C}}}^{m}$ near $0,$ for each $t\in(T-\epsilon,T),$ $L^{t}$ approximates a ‘Lawlor neck’ $L_{\boldsymbol{\phi}(t),A(t)}$ from Example 2.5, after a translation and a ${\rm U}(m)$ rotation in ${\mathbin{\mathbb{C}}}^{m}$. Here $A(t)>0$ is small and $A(t)\rightarrow 0$ as $t\rightarrow T,$ so that $L_{\boldsymbol{\phi}(t),A(t)}$ converges to a union $\Pi_{0}\cup\Pi_{\boldsymbol{\phi}(T)}$ of transversely intersecting special Lagrangian planes in ${\mathbin{\mathbb{C}}}^{m}$ as $t\rightarrow T$.

• (iv)

  For $t\in[T,T+\epsilon),$ there is a self-intersection point $p^{t}$ of $L^{t}$ where two local sheets $L^{t}_{\pm}$ of $L^{t}$ intersect transversely with $\mu_{L^{t}_{+},L^{t}_{-}}(p^{t})=1$. Here $p^{t},L^{t}_{\pm}$ depend smoothly on $t\in[T,T+\epsilon),$ with $p^{T}=p$.

• (v)

  We have $\theta_{L^{T}_{+}}(p^{T})=\theta_{L^{t}_{-}}(p^{T}),$ and $\theta_{L^{t}_{+}}(p^{t})<\theta_{L^{t}_{-}}(p^{t})$ for $t\in(T,T+\epsilon)$.

• (vi)

  The ${\mathbin{\mathbb{F}}}$-local systems $E^{t}$ for $t\in[T,T+\epsilon)$ are constructed from the ${\mathbin{\mathbb{F}}}$-local systems $E^{t^{\prime}}$ for $t^{\prime}\in(T-\epsilon,T)$ by deleting the ‘neck’ in $L^{t^{\prime}}$ and extending $E^{t^{\prime}}$ over $p^{t}$ in $L^{t}_{\pm}$ in the unique possible way (at least for $m\geqslant 3$).

• (vii)

  When $t\in[T,T+\epsilon),$ the bounding cochain $b^{t}$ for $L^{t}$ includes an element $b^{t}_{p^{t}}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ as in §2.6. This is of the form

  $$b^{t}_{p^{t}}=a_{0}P^{\lambda(t)}+\text{higher order terms,}$$

  where $a_{0}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)$ is the natural isomorphism induced from $E^{t^{\prime}}$ for $t^{\prime}\in(T-\epsilon,T)$ using (vi), and $\lambda(t)=\int_{T}^{t}\bigl(\theta_{L^{s}_{-}}(p^{s})-\theta_{L^{s}_{+}}(p^{s})\bigr){\rm d}s,$ so that $\lambda(T)=0$ and $\lambda(t)>0$ for $t\in(T,T+\epsilon)$ by (v).