Here is our suggestion for a programme to prove Conjecture 3.2 using Lagrangian MCF, building on Thomas and Yau [70]. We will state a conjecture about it in §3.9, after discussing issues that arise in the programme in §3.3–§3.8.

Define $Z:K_{0}(D^{b}{\mathbin{\mathscr{F}}}(M))\rightarrow{\mathbin{\mathbb{C}}}$ by (3.1), and define ${\mathbin{\cal P}}(\phi)$ for $\phi\in{\mathbin{\mathbb{R}}}$ to be the full subcategory of objects $A$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ isomorphic to $(L,E,b)$ for $L$ a (possibly singular) special Lagrangian of phase $e^{i\pi\phi}$ with $\theta_{L}=\pi\phi,$ as in Conjecture 3.2(c), or alternatively those objects $A$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ which for any $\epsilon>0$ are isomorphic to some $(L,E,b)$ with phase function $\theta_{L}:L\rightarrow(\pi\phi-\epsilon,\pi\phi+\epsilon),$ as in Conjecture 3.2(c$)^{\prime}$.

We must prove $(Z,{\mathbin{\cal P}})$ is a Bridgeland stability condition on $D^{b}{\mathbin{\mathscr{F}}}(M)$. We discuss only the problem of verifying Definition 3.1(iv) for objects $F=(L,E,b),$ where $(L,E)$ is a nonsingular, immersed Lagrangian brane with $HF^{*}$ unobstructed. For such $(L,E,b),$ we must construct a diagram

in $D^{b}{\mathbin{\mathscr{F}}}(M),$ where $L_{1},\ldots,L_{n}$ are either unique (possibly singular) special Lagrangians with $\theta_{L_{j}}=\pi\phi_{j}$ for $\phi_{1}>\phi_{2}>\cdots>\phi_{n},$ or else (possibly singular) Lagrangians with $\theta_{L_{j}}:L_{j}\rightarrow(\pi\phi_{j}-\epsilon,\pi\phi_{j}+\epsilon)$ for arbitrarily small $\epsilon>0$.

We aim to construct a unique family $\bigl\{(L^{t},E^{t},b^{t}):t\in[0,\infty)\bigr\}$ satisfying:

• (a)

  $(L^{0},E^{0},b^{0})=(L,E,b)$.

• (b)

  There is a (hopefully finite) series of singular times $0<T_{1}<T_{2}<\cdots,$ such that if $t\in[0,\infty)\setminus\{T_{1},T_{2},\ldots\}$ then $(L^{t},E^{t},b^{t})$ is an object in $D^{b}{\mathbin{\mathscr{F}}}(M)$ isomorphic to $(L,E,b),$ with $L^{t}$ a (possibly immersed or singular) compact, graded Lagrangian in $M,$ with $HF^{*}$ unobstructed.

• (c)

  The family $\bigl\{L^{t}:t\in[0,\infty)\setminus\{T_{1},T_{2},\ldots\}\bigr\}$ satisfies Lagrangian mean curvature flow, and $\bigl\{E^{t}:t\in[0,\infty)\setminus\{T_{1},T_{2},\ldots\}\bigr\}$ are locally constant in $t$. (As a shorthand, we will say that the family of Lagrangian branes $\bigl\{(L^{t},E^{t}):t\in[0,\infty)\setminus\{T_{1},T_{2},\ldots\}\bigr\}$ satisfies Lagrangian MCF.) The bounding cochains $b^{t}$ also change by a kind of ‘parallel transport’ for $t\in[0,\infty)\setminus\{T_{1},T_{2},\ldots\}$ as in §2.5–§2.6, to ensure that the isomorphism class of $(L^{t},E^{t},b^{t})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ remains constant.

• (d)

  Let $T_{i}$ for $i=1,2,\ldots$ be a singular time and $\epsilon>0$ be small, so that $\bigl\{L^{t}:t\in(T_{i}-\epsilon,T_{i})\bigr\}$ and $\bigl\{L^{t}:t\in(T_{i},T_{i}+\epsilon)\bigr\}$ satisfy Lagrangian MCF. As $t\rightarrow T_{i}$ in $(T_{i}-\epsilon,T_{i}),$ the flow usually undergoes a finite time singularity of Lagrangian MCF. But see §3.4 for a case in which the limit is smooth as $t\rightarrow T_{i}$ in $(T_{i}-\epsilon,T_{i}),$ and singular as $t\rightarrow T_{i}$ in $(T_{i},T_{i}+\epsilon)$.

  We do not require $(L^{T_{i}},E^{T_{i}},b^{T_{i}})$ to be an object in $D^{b}{\mathbin{\mathscr{F}}}(M),$ as the singularities of $L^{T_{i}}$ may be too bad, and if so, $b^{T_{i}}$ is meaningless.

  The topologies of $L^{t}$ for $t\in(T_{i}-\epsilon,T_{i}),$ and $L^{T_{i}},$ and $L^{t}$ for $t\in(T_{i},T_{i}+\epsilon),$ may all be different, so we may think of the (possibly singular) manifolds $L^{t}$ as undergoing a surgery at time $t=T_{i}$. Nonetheless, the family $\bigl\{L^{t}:t\in(T_{i}-\epsilon,T_{i}+\epsilon)\bigr\}$ is in a suitable sense continuous, for instance, as graded Lagrangian integral currents in $M$ in Geometric Measure Theory.

• (e)

  For the case of Conjecture 3.2(c), we have $\lim_{t\rightarrow\infty}L^{t}=L_{1}\cup\cdots\cup L_{n},$ where $L_{j}$ is a (possibly badly singular) special Lagrangian with phase $e^{i\pi\phi_{j}}$ and phase function $\theta_{L_{j}}=\pi\phi_{j},$ for $\phi_{1}>\phi_{2}>\cdots>\phi_{n}$. The local systems $E_{1},\ldots,E_{n},$ bounding cochains $b_{1},\ldots,b_{n}$ and morphisms in (3.4) are obtained from $\lim_{r\rightarrow\infty}E^{t}$ and $\lim_{t\rightarrow\infty}b^{t}$.

  For the case of Conjecture 3.2(c$)^{\prime}$, if $t\gg 0$ then there is a decomposition $L^{t}=L^{t}_{1}\amalg\cdots\amalg L^{t}_{n},$ such that $\theta_{L^{t}}$ maps $L^{t}_{j}\rightarrow(\pi\phi_{j}-\epsilon^{t},\pi\phi_{j}+\epsilon^{t})$ for $j=1,\ldots,n,$ where $\epsilon^{t}>0$ with $\epsilon^{t}\rightarrow 0$ as $t\rightarrow\infty$.