(i) In dimension $m>1$, Lagrangian MCF $\{L^{t}:t\in[0,\infty)\}$ starting from a compact, embedded Lagrangian $L^{0}$ can flow to immersed Lagrangians $L^{t}$ in finite time, as sketched in Figure 3.1, or vice versa. (When $m=1$, embedded curves remain embedded.)

Therefore, to carry out the programme above, we must include immersed Lagrangians in $D^{b}{\mathbin{\mathscr{F}}}(M)$, since otherwise in the situation of Figure 3.1 we could not continue the programme past $t=T$. This inclusion was discussed in §2.6, using the extension of [20] to immersed Lagrangians in Akaho and Joyce [2].

Observe that for Lagrangian MCF $\{L^{t}:t\in[0,T)\}$ of immersed, graded Lagrangians $L^{t}$ in a Calabi–Yau $m$-fold, the $L^{t}$ for $t\in[0,T)$ are all locally Hamiltonian isotopic in the sense of §2.6, but not necessarily globally Hamiltonian isotopic, as in Figure 3.1.

Thus, for immersed Lagrangian MCF we must deal with the possibility that even without finite time singularities, the flow may take us from Lagrangians with unobstructed $HF^{*}$ to Lagrangians with obstructed $HF^{*}$, or change the isomorphism class in $D^{b}{\mathbin{\mathscr{F}}}(M)$, since we explained in §2.6 that local Hamiltonian isotopies can do this. We discuss this further in §3.4.

In dimension $m=3$, I expect the programme above to be of comparable difficulty to the Poincaré Conjecture. As the dimension increases, so should the difficulty, as there will be more kinds of finite-time singularities to worry about.

We could modify the programme above by taking $L^{0}$ to be a generic Hamiltonian perturbation of $L$ in (a), rather than $L^{0}=L$. Then the Lagrangian MCF singularities occurring at the singular times $T_{1},T_{2},\ldots$ would have to have index 0. This might have the effect of limiting the kinds of singular Lagrangians that must be included in $D^{b}{\mathbin{\mathscr{F}}}(M)$ to make the programme work.

For similar ideas in MCF of hypersurfaces in ${\mathbin{\mathbb{R}}}^{n}$, see Angenent and Velázquez [6] who construct examples of non-generic finite time singularities of MCF, and Colding and Minicozzi [14], who classify the possible finite time singularities of MCF starting from a generic, compact, embedded surface $\Sigma^{2}$ in ${\mathbin{\mathbb{R}}}^{3}$.

It seems likely that the possible singularities occurring in $\lim_{t\rightarrow\infty}L^{t}$ may be too severe to incorporate as objects in $D^{b}{\mathbin{\mathscr{F}}}(M)$. Thus, although Conjecture 3.2(c) is more attractive, Conjecture 3.2(c$)^{\prime}$ is more plausible.