In Remark 3.8(i) we noted that Lagrangian MCF may take an immersed Lagrangian brane $(L^{t},E^{t})$ with $HF^{*}$ unobstructed to one $(L^{t^{\prime}},E^{t^{\prime}})$ for $t^{\prime}>t$ with $HF^{*}$ obstructed, without finite time singularities. This is a problem for the programme of §3.2, as we need $(L^{t},E^{t})$ to have $HF^{*}$ unobstructed for all $t$. We now discuss this problem in more detail, and explain how to solve it.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold and $\{(L^{t},E^{t}):t\in[0,T)\}$ a family of Lagrangian branes satisfying Lagrangian MCF. Suppose, for simplicity, that all the $L^{t}$ have transverse self-intersections. Then the self-intersection points of $L^{t}$ in $M$ depend smoothly on $t\in[0,T)$, so we can write $p^{t}$ for the intersection of local sheets $L^{t}_{+},L^{t}_{-}$ at $L^{t}$ for $t\in[0,T)$, where $p^{t},L^{t}_{\pm}$ depend smoothly on $t$. Then $\mu_{L^{t}_{+},L^{t}_{-}}(p^{t})$ is independent of $t$.

Suppose that $b^{t}$ is a bounding cochain for $L^{t}$ depending smoothly on $t$, with $(L^{t},E^{t},b^{t})\cong(L^{0},E^{0},b^{0})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$. Then $b^{t}$ evolves in time by a kind of ‘parallel transport’. Let $p^{t},L^{t}_{\pm}$ be as above with $\mu_{L^{t}_{+},L^{t}_{-}}(p^{t})=1$. Then as in §2.6, $b^{t}$ includes an element $b^{t}_{p^{t}}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$. The analysis of (2.18)–(2.21) holds, with $H^{t}=-\theta_{L^{t}}$. Thus, writing $b^{0}_{p^{0}}=\sum_{i=0}^{\infty}a_{i}P^{\lambda_{i}}$ with $a_{0}\neq 0$ and $0\leqslant\penalty\lambda_{0}<\lambda_{1}<\lambda_{2}<\cdots$, we have

and $b^{t}_{p^{t}}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}\subset\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}$, required for $b^{t}$ to be a bounding cochain, if and only if

We can now explain how Lagrangian MCF can flow from $HF^{*}$ unobstructed to $HF^{*}$ obstructed: as $t$ increases, we can cross a ‘wall’ at $t=T_{1}$ when the l.h.s. of (3.5) becomes negative, so that $b^{t}_{p^{t}}\notin\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ for $t>T_{1}$. Then $b^{t}$ is not a bounding cochain, and $(L^{t},E^{t})$ may have $HF^{*}$ obstructed.

To make this more explicit, let us simplify further, and suppose that $L^{t}$ has only two self-intersection points $p^{t},q^{t}$ with $\mu_{L^{t}_{+},L^{t}_{-}}(p^{t})=1$ and $\mu_{L^{t}_{+},L^{t}_{-}}(q^{t})=2$, and there are only two $J$-holomorphic curves $\Sigma^{t}_{1},\Sigma^{t}_{2}$ with boundary in $L^{t}$ which are relevant to obstructions to $HF^{*}$, which are as shown in Figure 3.2, so that $\Sigma^{t}_{1}$ has two corners at $p^{t},q^{t}$ and $\Sigma^{t}_{2}$ one corner at $q^{t}$. Note that $\Sigma^{t}_{2}$ is the type of curve in Figure 2.3 that can cause obstructions to immersed $HF^{*}$.

Then $(L^{t},E^{t})$ has $HF^{*}$ unobstructed if and only if $\mathop{\rm area}(\Sigma^{t}_{2})\geqslant\mathop{\rm area}(\Sigma^{t}_{1})$, and if so, the bounding cochain $b^{t}$ has

where $0\neq a_{0}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p^{t}},E_{-}^{t}|_{p^{t}}\bigr)$. We can think of $\Sigma^{t}_{2}-\Sigma^{t}_{1}$ as a ‘virtual $J$-holomorphic curve’ with ‘virtual area’ $\mathop{\rm area}(\Sigma^{t}_{2})-\mathop{\rm area}(\Sigma^{t}_{1})$ and one corner at $p^{t}$, which obstructs $HF^{*}$ if this virtual area is negative.

Under Lagrangian MCF we have

Suppose now that the family $\{(L^{t},E^{t}):t\in[0,T)\}$ passes from $HF^{*}$ unobstructed when $t<T_{1}$ to $HF^{*}$ obstructed when $t>T_{1}$. Then $\mathop{\rm area}(\Sigma^{t}_{2})-\mathop{\rm area}(\Sigma^{t}_{1})$ crosses zero at $t=T_{1}$ going from positive to negative, so (3.7) shows that

We claim that in the programme of §3.2, the correct thing to do is to change $L^{t}$ for $t\geqslant T_{1}$ by doing a surgery at $p^{t}$ when $t=T_{1}$, a Lagrangian connected sum of the two sheets $L^{t}_{+},L^{t}_{-}$ at $p^{t}$, so that $L^{t}$ for $T_{1}<t<T_{1}+\epsilon$ looks roughly like Figure 3.3. We will call this surgery ‘opening a neck’. The self-intersection $p^{t}$ is

now gone, and there are two $J$-holomorphic discs $\Sigma^{t}_{1},\Sigma^{t}_{2}$ with one corner at $q^{t}$. Since we do the surgery when $\mathop{\rm area}(\Sigma^{t}_{1})=\mathop{\rm area}(\Sigma^{t}_{2})$, we have $\mathop{\rm area}(\Sigma^{t}_{1})=\mathop{\rm area}(\Sigma^{t}_{2})$ for all $t>T_{1}$, though $\Sigma^{t}_{1},\Sigma^{t}_{2}$ are in different relative homology classes. As their areas are equal, the obstructions from $\Sigma^{t}_{1},\Sigma^{t}_{2}$ cancel for suitable $E^{t}$, and $(L^{t},E^{t})$ for $t>T_{1}$ has $HF^{*}$ unobstructed.

We have $\mu_{L_{+}^{T_{1}},L_{-}^{T_{1}}}(p^{T_{1}})=1$ and $\theta_{L_{+}^{T_{1}}}(p^{T_{1}})\geqslant\theta_{L_{-}^{T_{1}}}(p^{T_{1}})$ by (3.8). Suppose strict inequality holds, $\theta_{L_{+}^{T_{1}}}(p^{T_{1}})>\theta_{L_{-}^{T_{1}}}(p^{T_{1}})$. Then from Definition 2.20, we see that there is an identification $T_{p^{T_{1}}}M\cong{\mathbin{\mathbb{C}}}^{m}$ identifying $J|_{p^{T_{1}}},g|_{p^{T_{1}}}$ with the standard versions on ${\mathbin{\mathbb{C}}}^{m}$, and identifying $T_{p^{T_{1}}}L_{+}^{T_{1}},T_{p^{T_{1}}}L_{-}^{T_{1}}$ with the Lagrangian planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}$ in (2.12) for $\phi_{1},\ldots,\phi_{m}\in(0,\pi)$ with $0<\phi_{1}+\cdots+\phi_{m}<\pi$, where $\phi_{1}+\cdots+\phi_{m}<\pi$ comes from $\mu_{L_{+}^{T_{1}},L_{-}^{T_{1}}}(p^{T_{1}})=1$ and $\theta_{L_{+}^{T_{1}}}(p^{T_{1}})>\theta_{L_{-}^{T_{1}}}(p^{T_{1}})$.

Thus, by Example 2.13 there is a unique, exact Joyce–Lee–Tsui Lagrangian MCF expander $L_{\boldsymbol{\phi}}^{1}$ with $\alpha=1$ in $T_{p^{T_{1}}}M$ asymptotic to $T_{p^{T_{1}}}L_{+}^{T_{1}}\cup T_{p^{T_{1}}}L_{-}^{T_{1}}$, and Theorem 2.14 shows that $L_{\boldsymbol{\phi}}^{1}$ is the only LMCF expander with $\alpha=1$ in $T_{p^{T_{1}}}M$ asymptotic to $T_{p^{T_{1}}}L_{+}^{T_{1}}\cup T_{p^{T_{1}}}L_{-}^{T_{1}}$. Note that $\sqrt{2(t-T_{1})}\cdot L_{\boldsymbol{\phi}}^{1}$ for $t>T_{1}$ satisfy Lagrangian MCF in $T_{p^{T_{1}}}M\cong{\mathbin{\mathbb{C}}}^{m}$. We now aim to define the $L^{t}$ for $T_{1}<t<T_{1}+\epsilon$ by gluing in $\sqrt{2(t-T_{1})}\cdot L_{\boldsymbol{\phi}}^{1}$ into $L^{T_{1}}$ near $p^{T_{1}}$.

To define the local systems $E^{t}$ for $t>T_{1}$, note that $b^{T_{1}}_{p^{T_{1}}}=a_{0}+\cdots$ by (3.6), where $0\neq a_{0}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{T_{1}}|_{p^{T_{1}}},E_{-}^{T_{1}}|_{p^{T_{1}}}\bigr)$. As $E^{T_{1}}$ has rank one, $a_{0}\neq 0$ implies that $a_{0}$ is an isomorphism. For $T_{1}<t<T_{1}+\epsilon$, we define $E^{t}$ to be equal to $E^{T_{1}}$ away from the ‘neck’ region joining $L_{+}^{T_{1}}$ with $L_{-}^{T_{1}}$, and on the ‘neck’ region we use the isomorphism $a_{0}$ to identify $E^{T_{1}}|_{L^{+}}$ and $E^{T_{1}}|_{L^{-}}$. This choice of $E^{t}$ is necessary for the obstructions to $HF^{*}$ for $(L^{t},E^{t})$ from $\Sigma_{1}^{t},\Sigma_{2}^{t}$ to cancel.

The bounding cochain $b^{t}$ for $T_{1}<t<T_{1}+\epsilon$ should be roughly equal to $b^{T_{1}}$ away from the ‘neck’ region. On the ‘neck’ region, $b^{t}$ should somehow encode the higher order terms in $b^{T_{1}}_{p^{T_{1}}}=a_{0}+\cdots$, possibly in the form $b^{t}\approx\log\bigl(a_{0}^{-1}\circ b^{T_{1}}_{p^{T_{1}}}\bigr)\cdot[{\mathbin{\cal S}}^{m-1}_{t}]$, where $[{\mathbin{\cal S}}^{m-1}_{t}]\in C_{m-1}(L^{t},{\mathbin{\mathbb{Z}}})$ is a fundamental cycle for the new small $(m\!-\!1)$-sphere ${\mathbin{\cal S}}^{m-1}_{t}$ spanning the ‘neck’ in $L^{t}$.

Our discussion has shown the following rather neat:

Suppose that $(L^{t},E^{t})$ has $HF^{*}$ unobstructed for $0\leqslant\penalty t<T_{1}<T,$ but at $t=T_{1}$ crosses a ‘wall’ into $HF^{*}$ obstructed, because at a transverse self-intersection point $p$ of $L^{T_{1}}$ with $\mu_{L_{+}^{T_{1}},L_{-}^{T_{1}}}(p)=1,$ the data $b_{p}^{t}$ in the bounding cochain $b^{t}$ leaves $\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p},E_{-}^{t}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ in $\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{+}^{t}|_{p},E_{-}^{t}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}$ when $t=T_{1}$.

Then (at least if strict inequality holds in (3.8)) there is a unique Lagrangian MCF expander in $T_{p}M$ asymptotic to $T_{p}L_{+}^{T_{1}}\cup T_{p}L_{-}^{T_{1}},$ which we can (conjecturally) use to do a surgery at $t=T_{1}$ so that the flow can continue for $t>T_{1}$ with $HF^{*}$ unobstructed, as in §3.2. The analogue does not hold for flowing from $HF^{*}$ obstructed to $HF^{*}$ unobstructed.

It also suggests a research project:

In the next example we use ‘opening necks’ to resolve an apparent counterexample to our programme.