Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold, either compact or suitably convex at infinity, and $D^{b}{\mathbin{\mathscr{F}}}(M)$ an enlarged version of the derived Fukaya category of Lagrangian branes in $M$ from [20], including classes of immersed or singular Lagrangians, depending on the dimension $m$:

• (i)

  When $m=1,$ $D^{b}{\mathbin{\mathscr{F}}}(M)$ can be the usual derived Fukaya category of nonsingular, embedded Lagrangian branes.

• (ii)

  When $m\geqslant 2,$ $D^{b}{\mathbin{\mathscr{F}}}(M)$ must include immersed Lagrangians, as in Akaho and Joyce [2] and §2.6. For $m=2,$ these are all of $D^{b}{\mathbin{\mathscr{F}}}(M)$.

• (iii)

  When $m\geqslant 3,$ $D^{b}{\mathbin{\mathscr{F}}}(M)$ must also include singular Lagrangians with stable special Lagrangian singularities, as in §3.6. When $m=3,$ these include Lagrangians with isolated conical singularities in the sense of [32, 33, 34, 35, 36] modelled on the special Lagrangian $T^{2}$-cone from (2.4), and this may be the only kind of stable singularity when $m=3$. When $m\geqslant 4,$ stable singularities may be more complicated, and need not be isolated.

Let $(L,E)$ be a Lagrangian brane in $M$ with $HF^{*}$ unobstructed, and suppose $L$ is generic in its Hamiltonian isotopy class. Let $b$ be a bounding cochain for $(L,E)$. Then there is 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 finite series of singular times $0<T_{1}<T_{2}<\cdots<T_{N}$ such that if $t\in[0,\infty)\setminus\{T_{1},\ldots,T_{N}\}$ 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 $(M,\omega),$ with $HF^{*}$ unobstructed.

• (c)

  The family $\bigl\{L^{t}:t\in[0,\infty)\setminus\{T_{1},\ldots,T_{N}\}\bigr\}$ satisfies Lagrangian mean curvature flow, and $\bigl\{E^{t}:t\in[0,\infty)\setminus\{T_{1},\ldots,T_{N}\}\bigr\}$ is locally constant in $t$. 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)

  At each singular time $T_{1},\ldots,T_{N},$ the flow undergoes a surgery, which may involve a finite time singularity of Lagrangian MCF, and a change in the topology of $L^{t}$. The kinds of surgery allowed include ‘opening a neck’ as in §3.4 when $m\geqslant 1,$ ‘neck pinches’ as in §3.5 when $m\geqslant 2,$ transitions to and from Lagrangians with ‘stable special Lagrangian singularities’ as in §3.6 when $m\geqslant 3,$ and ‘collapsing zero objects’ as in §3.7 for $m\geqslant 1$ (the latter is excluded for almost calibrated Lagrangians).

  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.

• (e)

  The family $\{L^{t}:t\in[0,\infty)\}$ is continuous as graded Lagrangian integral currents in $M$ in Geometric Measure Theory.

  In graded Lagrangian integral currents, we have $\lim_{t\rightarrow\infty}L^{t}=L_{1}+\cdots+L_{n}$ for some $n\geqslant 0,$ where $0\neq L_{j}$ for $j=1,\ldots,n$ is a nonzero, compactly-supported, graded, special Lagrangian integral current with phase $e^{i\pi\phi_{j}}$ and grading $\theta_{L_{j}}=\pi\phi_{j},$ with $\phi_{1}>\cdots>\phi_{n}$.

  For the Bridgeland stability condition $(Z,{\mathbin{\cal P}})$ on $D^{b}{\mathbin{\mathscr{F}}}(M)$ discussed in Conjecture 3.2, if $n=1$ then $(L,E,b)\in{\mathbin{\cal P}}(\phi_{1}),$ and otherwise $(L,E,b)\notin{\mathbin{\cal P}}(\phi)$ for any $\phi\in{\mathbin{\mathbb{R}}}$.