Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold, and $(L_{1},E_{1}),(L_{2},E_{2})$ be embedded, transversely-intersecting, special Lagrangian branes in $M$ with phases $e^{i\pi\phi_{1}},e^{i\pi\phi_{2}}$ for $\phi_{1}<\phi_{2}$, with $HF^{*}$ unobstructed. Choose bounding cochains $b_{1},b_{2}$ for $(L_{1},E_{1}),(L_{2},E_{2})$. Let $0\!\neq\!\beta\!\in\!HF^{1}\bigl((L_{2},E_{2},b_{2}),(L_{1},E_{1},b_{1})\bigr)$, and $(\beta_{p})\in CF^{1}\bigl((L_{2},E_{2}),(L_{1},E_{1})\bigr)$ represent $\beta$, where for all $p\in L_{1}\cap L_{2}$ with $\mu_{L_{2},L_{1}}(p)=1$ we have $\beta_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}|_{p},E_{1}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}$ .

Suppose $\beta_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}|_{p},E_{1}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ for all $p$. Set $(L,E)=(L_{1},E_{1})\cup(L_{2},E_{2})$, considered as an immersed Lagrangian brane in $M$. Then using the notation of §2.6, $b=b_{1}\oplus b_{2}\oplus(\beta_{p})$ is a bounding cochain for $(L,E)$, where $b_{\rm ch}=b_{1}\oplus b_{2}$ in $C_{m-1}(L,\Lambda_{\rm nov}^{+})=C_{m-1}(L_{1},\Lambda_{\rm nov}^{+})\oplus C_{m-1}(L_{2},\Lambda_{\rm nov}^{+})$, and the data $b_{p}$ for each $p\in M$ at which two local sheets $L_{+},L_{-}$ of $L$ intersect transversely with $\mu_{L_{+},L_{-}}(p)=1$ are $b_{p}=\beta_{p}$ if $L_{+}=L_{2}$, $L_{-}=L_{1}$, and $b_{p}=0$ otherwise. We now have a distinguished triangle in the derived Fukaya category $D^{b}{\mathbin{\mathscr{F}}}(M)$ of immersed Lagrangians

Let us apply the programme of §3.2 to $(L,E,b)$. Since $L$ is a union of special Lagrangians of different phases, it is stationary under immersed Lagrangian MCF, so the obvious answer is that $(L^{t},E^{t},b^{t})=(L,E,b)$ for all $t\in[0,\infty)$. Equation (3.9) gives a diagram for $(L,E,b)$ of the form (3.4) with $n=2$

However, in §3.2 we want such a diagram with $\phi_{1}>\phi_{2}$, but we assume that $\phi_{1}<\phi_{2}$. So writing $(L^{t},E^{t},b^{t})=(L,E,b)$ for all $t\in[0,\infty)$ does not satisfy the programme of §3.2, as although we have long-time existence of Lagrangian MCF, the limiting behaviour at infinity is wrong, and this looks like a counterexample.

Here is the explanation. Although (at least initially) the $L^{t},E^{t}$ are independent of $t$, the bounding cochains $b^{t}$ do evolve in time. Suppose $p\in L_{1}\cap L_{2}$ with $\mu_{L_{2},L_{1}}(p)=1$. Then (2.18)–(2.21) with $H_{L_{j}}=-\theta_{L_{j}}=-\pi\phi_{j}$ for $j=1,2$ shows that the data $b_{p}^{t}$ in $b^{t}$ should evolve according to the equation

so as $b_{p}^{0}=\beta_{p}$, the solution is $b_{p}^{t}=P^{\pi(\phi_{1}-\phi_{2})t}\cdot\beta_{p}$. Thus, we have

at least for small $t$. Write $\beta_{p}=a_{p}P^{\lambda_{p}}+\cdots$ if $\beta_{p}\neq 0$, where $0\neq a_{p}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}|_{p},E_{1}|_{p}\bigr)$ and $\lambda_{p}\geqslant 0$, and set $\lambda_{p}=\infty$ if $\beta_{p}=0$. Then $b_{p}^{t}=a_{p}P^{\lambda_{p}+\pi(\phi_{1}-\phi_{2})t}+\cdots$, so $b_{p}^{t}\in\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}^{t}|_{p},E_{1}^{t}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ if $t\in[0,\lambda_{p}/\pi(\phi_{2}-\phi_{1})]$.

Thus, at time $T=(\min_{p}\lambda_{p})/\pi(\phi_{2}-\phi_{1})$, the flow crosses a ‘wall’ after which $b^{t}$ in (3.10) is no longer a bounding cochain, as $b_{p}^{t}$ leaves $\mathop{\rm Hom}\nolimits_{\mathbin{\mathbb{F}}}\bigl(E_{2}^{t}|_{p},E_{1}^{t}|_{p}\bigr)\otimes_{\mathbin{\mathbb{F}}}\Lambda_{\rm nov}^{\geqslant 0}$ for some $p$. We claim that the right thing to do is to ‘open a neck’ at time $t=T$ at each $p$ with $\lambda_{p}$ minimal, gluing in a Joyce–Lee–Tsui LMCF expander. Then $L^{t},E^{t}$ will undergo some nontrivial evolution for $t>T$.

To see that a suitable LMCF expander exists to glue in at $p$, note that $\theta_{L_{j}}(p)=\pi\phi_{j}$ for $j=1,2$, so $\theta_{L_{1}}(p)<\theta_{L_{2}}(p)$ by assumption, and as $\mu_{L_{2},L_{1}}(p)=1$, the first equation of (2.13) gives $\theta_{L_{2}}(p)<\theta_{L_{1}}(p)+\pi$. These are the conditions for the existence of an LMCF expander in $T_{p}M$ asymptotic to $T_{p}L_{1}\cup T_{p}L_{2}$.