(i) The enlargement of $D^{b}{\mathbin{\mathscr{F}}}(M)$ envisaged in (c),(c$)^{\prime}$ adds more objects to $D^{b}{\mathbin{\mathscr{F}}}(M)$, but it need not change $D^{b}{\mathbin{\mathscr{F}}}(M)$ up to equivalence.

An example of the kind of enlargement the author has in mind is including immersed Lagrangians in $D^{b}{\mathbin{\mathscr{F}}}(M)$, as in §2.6. We have embedded and immersed derived Fukaya categories $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm em}\subset D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm im}$, but if every immersed Lagrangian $(L,E,b)$ in $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm im}$ is equivalent to a twisted complex of embedded Lagrangians, then $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm em}\simeq D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm im}$.

For many applications in symplectic topology, one only really cares about $D^{b}{\mathbin{\mathscr{F}}}(M)$ up to equivalence, so adding extra geometric objects to $D^{b}{\mathbin{\mathscr{F}}}(M)$ in this way is unnecessary. But for Conjecture 3.2(c),(c$)^{\prime}$, it is vital — if an isomorphism class in ${\mathbin{\cal P}}(\phi)$ contains a unique special Lagrangian representative $(L,E,b)$, and $L$ happens to be immersed, then restricting to embedded Lagrangians would make Conjecture 3.2(c) false. Similarly, we will see that the programme of long-time existence for Lagrangian MCF we outline below must take place in an enlarged category of Lagrangians to have any chance of working.

Note however that Thomas and Yau’s method does not exclude the possibility that $L^{\prime}\rightarrow L$ and $L^{\prime\prime}\rightarrow L$ are non-isomorphic $k$-fold multiple covers of a non-simply-connected special Lagrangian $L$ in $M$ for $k>1$, with $(L^{\prime},E^{\prime},b^{\prime})\cong(L^{\prime\prime},E^{\prime\prime},b^{\prime\prime})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$. A good uniqueness statement in Conjecture 3.2(c) may be that the special Lagrangian integral current in Geometric Measure Theory induced by $L$ is unique, so that in the case above the special Lagrangian integral currents of both $L^{\prime},L^{\prime\prime}$ would be $kL$.

Kontsevich’s Homological Mirror Symmetry Conjecture [44] roughly says that Calabi–Yau $m$-folds should exist in ‘mirror pairs’ $M,\check{M}$ for which there should be an equivalence of triangulated categories

where $D^{b}\mathop{\rm coh}(\check{M})$ is the derived category of coherent sheaves on $\check{M}$. (Really $\check{M}$ should be defined over the Novikov ring $\Lambda_{\rm nov}$.)

Kontsevich [44] proved (3.2) when $M$ is an elliptic curve (a Calabi–Yau 1-fold). Seidel [63] proved it for $M$ a quartic surface in ${\mathbin{\mathbb{CP}}}^{3}$ (a Calabi–Yau 2-fold), and Sheridan [65] proved it for $M$ a smooth Calabi–Yau $m$-fold hypersurface in ${\mathbin{\mathbb{CP}}}^{m+1}$ for $m\geqslant 3$. If (3.2) holds then stability conditions on $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ are equivalent to stability conditions on $D^{b}\mathop{\rm coh}(\check{M})$. But derived categories of coherent sheaves are generally better understood than derived Fukaya categories.

Bridgeland stability conditions on $D^{b}\mathop{\rm coh}(M)$ are defined by Bridgeland [10, Ex. 5.4] for $M$ a Calabi–Yau 1-fold and [11] for $M$ an algebraic $K3$ surface (a Calabi–Yau 2-fold). Assuming a conjecture on ‘Bogomolov–Gieseker type inequalities’, Bayer, Macrì and Toda [7] construct Bridgeland stability conditions on $D^{b}\mathop{\rm coh}(M)$ for $M$ a Calabi–Yau 3-fold; the conjecture is proved by Macioca and Piyaratne [48, 49] when $M$ is an abelian 3-fold.

Combining the two, one may be able to construct examples of Bridgeland stability conditions on $D^{\pi}{\mathbin{\mathscr{F}}}(M)$ for $M$ a Calabi–Yau 1-fold, 2-fold or 3-fold.