In order to understand limits of mean curvature flow it will be useful to have the following concept; an analogue for Lagrangians of the Jordan-Hölder filtration of sheaves (see [HuL] 1.5, for instance).

In sheaf theory the Jordan-Hölder filtration need not be unique, but the decomposition is. For smooth connected Lagrangians, with connected $L_{i}$ for all $i$, however, we expect the filtration to be unique too; the difference is essentially that while direct sum is an operation on bundles, we are proposing that its mirror is the (singular) union of Lagrangians, and this cannot give a smooth Lagrangian if there is non-zero Floer cohomology between the two Lagrangians.

If we assume the conjecture of [Th] and Section 7, and the properties of Floer cohomology [FO3] for all of the above Lagrangians (e.g. if they are homology spheres), we can demonstrate the existence and uniqueness of the Jordan-Hölder filtration for a Lagrangian $L$ whose phase function of $L$ satisfies $\sup\theta_{L}-\inf\theta_{L}<\pi$.

Without loss of generality we may assume (by rotating $\Omega$) that $\theta$ lies between $\pi/2-\epsilon$ and $-\pi/2+\epsilon$, for some $\epsilon>0$. By the inequality (5.2) above, then, any $L_{1}\#L^{\prime}$ destabilising it will satisfy $\phi(L_{1}),\phi(L^{\prime})\in(-\pi/2+\epsilon,\pi/2-\epsilon)$.

We choose such an $L_{1}$ of maximal phase, and, amongst other such $L_{1}$s of the same phase, minimal $\int_{L_{1}}\mathrm{Re}\,\Omega$ (for the purposes of this proof we will call this quantity cohomological volume). This still need not specify $L_{1}$ uniquely though.

We claim that such an $L_{1}$ must be stable by construction. Any subobject of $l\leq L_{1}$ would also be a subobject of $L$ and so by the construction of $L_{1}$ must either have smaller phase, which is not possible since it destabilises $L_{1}$, or equal phase and greater or equal cohomological volume. But $L_{1}=l\#l^{\prime}$, where $l^{\prime}=L_{1}/l$ has phase $\phi(l^{\prime})=\phi(L_{1})\in(-\pi/2,\pi/2)$ and so positive cohomological volume $\int_{l^{\prime}}\mathrm{Re}\,\Omega$. So the complex numbers

all have positive real part, so that the cohomological volume of $l$ is strictly less that that of $L_{1}$, a contradiction.

We then apply the same procedure to $L^{\prime}$, producing an $L_{2}\hookrightarrow L^{\prime}$, and so on. By construction $\phi(L^{\prime}/L_{2})\leq\phi(L^{\prime})\leq\phi(L)<\pi/2-\epsilon$, and there is a canonical morphism (5.1) in $HF^{0}(L,L^{\prime}/L_{2})\neq 0$, making $\phi(L^{\prime}/L_{2})\geq\inf_{L}\theta_{L}>-\pi/2+\epsilon$ by (5.2).

Thus, inductively, we get the same inequalities at each stage, and the cohomological volume of $L^{\prime}$ decreases strictly with each decomposition $L\approx L_{1}\#\ldots\#L_{n}\#L^{\prime}$. The cohomological volume of any $l$ with phase $\phi(l)\in(-\pi/2+\epsilon,\pi/2-\epsilon)$ is greater than (or equal to in the SLag case) $\cos(\pi/2-\epsilon)\int_{l}\vol$, by (1.1), where $\vol$ is its Riemannian volume form. This is bounded below above zero, so the process can have at most a finite number of steps.

This gives us the Jordan-Hölder filtration; next we consider uniqueness when the $L_{i}$s are connected (assuming the conjectures of [Th] and Section 7 and some Floer cohomology). Suppose that $L_{1}^{\prime}\leq L_{2}^{\prime}\leq\ldots\leq L$ is another such connected decomposition. If $HF^{0}(L_{1}^{\prime},L_{1})\neq 0$ then by the proof of Theorem 4.3 (which applies as $L_{1}^{\prime}$ and $L_{1}$ have the same phase), $L_{1}$ and $L_{1}^{\prime}$ are equal, and we pass to $L_{2}$.

If, however, $HF^{0}(L_{1}^{\prime},L_{1})=0$, then we claim that $HF^{0}(L_{1}^{\prime},L/L_{1})\neq 0$. Again this should follow from standard facts about Floer cohomology, in particular a long exact sequence $HF^{*}(L_{1}^{\prime},L_{1})\to HF^{*}(L_{1}^{\prime},L)\to HF^{*}(L_{1}^{\prime},L/L_{1})\to HF^{*+1}(L_{1}^{\prime},L_{1})$. For $L/L_{1}$ a sphere this is Seidel’s exact sequence ([S3] Theorem 3.3), and in general one can establish it at the level of chains by good choices of hamiltonian perturbations as in Section 5.2; as usual the problem is in controlling the differential, i.e. holomorphic discs.

Assuming this we may pass to $L_{2}$; inductively we eventually obtain that $L_{1}^{\prime}$ is isomorphic to one of the graded pieces $L_{i+1}/L_{i}$ of the original filtration, and is a subobject of $L_{i+1}$ but not of $L_{i}$. But this gives us a contradiction (in contrast to the sheaf analogue), since we have that both $L_{i+1}\approx L_{i}\#L_{1}^{\prime}$ and $L_{1}^{\prime}$ is a subobject of $L_{i+1}$. The first condition ensures that there are representatives of the hamiltonian deformation classes such that $L_{i+1}$ and $L_{1}^{\prime}$ have no index $n$ intersection points by the construction of (5.1), so that $HF^{n}(L_{i+1},L_{1}^{\prime})=0$. But this is $HF^{0}(L_{1}^{\prime},L_{i+1})^{*}$, which cannot vanish by the second condition. (It is here we use the connectivity condition, i.e. that the connect sum $L_{i+1}=L_{1}^{\prime}\#(L_{i+1}/L_{1}^{\prime})$ is not a trivial disjoint union. Without the connectivity condition the usual proof (e.g. [HuL] 1.5) the the Jordan-Hölder decomposition (rather than filtration) of sheaves is unique applies to Lagrangians, now that we have proved or assumed all (the mirror analogues of) the algebraic facts used for sheaves in terms of Floer cohomology instead of Exts.)

So in the simplest case of instability, such as the example of Joyce considered in [Th], where $L=L_{1}\#L_{2}$ is the only relevant decomposition of $L$ with $\phi(L_{1})\geq\phi(L_{2})$, the Jordan-Hölder decomposition (5.4) would be simply $L_{1}\cup L_{2}$ (where the $L_{i}$ are SLag representatives of their classes). This, like all such decompositions, is in the closure of the hamiltonian deformation orbit of $L$ while not being in the orbit itself.

This should have relevance to the Schoen-Wolfson programme [SW1], [SW2] to find canonical representatives (in a fixed hamiltonian deformation class) of Lagrangian homology classes using volume minimisers and so SLags (they do not use a flow, but regularity results to study minimising currents). Our conjecture (as in [Th] and later in Section 7) should either provide a unique SLag in a hamiltonian deformation class, or a number of SLags in a Jordan-Hölder decomposition.

For instance in the example above of $L_{1}\#L_{2}$ in 2 dimensions (where their programme has been worked out [SW2]) we would produce SLags in the classes of $L_{1}$ and $L_{2}$, but we could also form $L_{2}\#L_{1}$; this could then be stable (it is no longer destabilised by either of the $L_{i}$; if the phases of the $L_{i}$ are sufficiently close one can show that in fact nothing else destabilises it either) and we should recover a SLag in this class (and so in the same homology class in two dimensions).

Since in two dimensions SLags are just holomorphic curves with respect to a different complex structure, this places heavy restrictions on stability. Take the $L_{i}$ above to be spheres in $K3$ surfaces. Then any holomorphic sphere is unique in its homology class (it has negative self intersection $-2$, so does not lie in a pencil). Any other hamiltonian deformation class must therefore be unstable. Good examples are provided by taking a stable (SLag/holomorphic) sphere, and applying the square of a Dehn twist $T^{2}_{L_{1}}$ to it; this preserves homology classes but can change hamiltonian deformation classes. If it does it should produce an unstable Lagrangian with copies of $L_{1}$ in its Jordan-Hölder decomposition; this happens in all simple cases. $L_{1}\#L_{2}$ is taken to $L_{2}\#L_{1}$, for instance; only one of these can be stable, the other having a Jordan-Hölder decomposition $L_{1}\cup L_{2}$ in the simplest case.

More generally, instead of studying the action on individual (S)Lags of symplectomorphisms like $T^{2}_{L_{1}}$ above, we could try to study them all at once by studying the Lagrangian graph of the symplectomorphism in $X\times X$, and its mean curvature flow. This looks for minimal energy representatives of the hamiltonian isotopy class of a symplectomorphism, and breaks graphs up into correspondences representing singular maps (birational maps in the hyperkähler case) with singularities concentrated in loci whose stability is affected by the symplectomorphism. For a Dehn twist $T_{L}$, for instance, we would expect to get the graph $\Delta$ of the identity union $L\times L$. This also shows what the analogue of a Dehn twist $T_{L}$ should be when $L$ is not a sphere but a rational homology sphere (so that it is still spherical to complex coefficients, and so mirror to a twist on the derived category of sheaves on the mirror Calabi-Yau [ST]). Namely $\Delta\cup L\times L$ is a Lagrangian correspondence in $X\times X$ which should give an automorphism of the derived Fukaya category of $X$ (by the usual Fourier-Mukai-type construction) not induced by a symplectomorphism of $X$.