Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold for $m\geqslant 2$, $L$ a compact Lagrangian in $M$, and $p\in L$. In [57], Neves defines another Lagrangian $\tilde{L}$ in $M$, which is Hamiltonian isotopic to $L$ and coincides with $L$ except in a small open neighbourhood of $p$. Here $L,\tilde{L}$ are locally $\mathop{\rm SO}\nolimits(m)$ surfaces of revolution on the curves in sketched in Figure 3.7. (Actually Neves restricts to $m=2$, but the same ideas should work for all $m\geqslant 2$.)

Neves’ main result [57, Th. A] is that Lagrangian MCF starting from $\tilde{L}$ develops finite time singularities. This is important, as it shows that finite time singularities in Lagrangian MCF are unavoidable in many situations (although note that $\tilde{L}$ has phase variation greater than $\pi$, so this does not show that almost calibrated Lagrangian MCF has finite time singularities).

What actually happens in Lagrangian MCF starting from $\tilde{L}$? Neves’ proof does not tell us, as he assumes for a contradiction that no finite time singularity occurs. The author expects a Lagrangian MCF with surgeries $\{L^{t}:t\in[0,T)\}$ in $M$ with $L^{0}=\tilde{L}$, with two singular times $0<T_{1}<T_{2}<T$. For $t\in[0,T_{1}),$ $L^{t}$ looks much like $\tilde{L}$, but as $t\rightarrow T_{1}$ in $[0,T_{1})$, the region marked with crosses ‘$\times$’ in Figure 3.7 undergoes a ‘neck pinch’. At $t=T_{1}$, as sketched in Figure 3.8, $L^{T_{1}}$ decomposes as $L^{T_{1}}_{1}\amalg L^{T_{1}}_{2}$, where $L^{T_{1}}_{1}$ is a small immersed ${\mathbin{\cal S}}^{m}$ near $p$ with one transverse self-intersection point with $\mu_{L_{+},L_{-}}(p)=m+1$, a ‘Whitney sphere’ as in Example 3.23, and $L^{T_{1}}_{2}$ looks quite like the original $L$.

Then as $t$ increases from $T_{1}$ to $T_{2}$, the component $L^{t}_{1}$ should shrink to a point, until at the second singular time $t=T_{2}$ it undergoes ‘collapsing a zero object’ as in Principle 3.25. Meanwhile, the Lagrangian MCF of $L^{t}_{2}$ looks quite like that of the original $L$, and continues for $t>T_{2}$. Thus, at least conjecturally, Neves’ examples [57] are not counterexamples to the programme of §3.2.