Provided one can prove long time existence of LMCF, the total mass of $L_{t}$ will be uniformly bounded since it decreases during the flow. Under mild conditions to ensure $L_{t}$ does not escape to spatial infinity (e.g. if the ambient Calabi-Yau manifold is compact), one can extract the infinite time subsequential limits of $L_{t}$ as currents. From the heat equation (51) on the Lagrangian angle $\theta$,

$$(\partial_{t}-\Delta_{L_{t}})|\theta|^{2}=-2|\nabla\theta|^{2}=-2|\vec{H}|^{2}.$$

If the Lagrangians remain sufficiently smooth, then an integration by part calculation shows

$$\int_{0}^{T}\int_{L_{t}}|\vec{H}|^{2}dvol_{L_{t}}dt=\frac{1}{2}\left(\int_{L_{0}}|\theta|^{2}dvol_{L_{0}}-\int_{L_{T}}|\theta|^{2}dvol_{L_{T}}\right).$$

Even if the volume mass can jump down at discrete time, such as during the collapsing of zero objects, we still expect

$$\int_{0}^{T}\int_{L_{t}}|\vec{H}|^{2}dvol_{L_{t}}dt\leq\frac{1}{2}\int_{L_{0}}|\theta|^{2}dvol_{L_{0}}<\infty,\quad\forall T>0.$$

(53)

In particular we can find a sequence of time $t_{i}\to\infty$, with

$$\int_{L_{t_{i}}}|\nabla\theta|^{2}dvol\to 0.$$

This strongly suggests that the subsequential limit is a union of special Lagrangian currents with multiplicities.

In the heursitic logic of Thomas-Yau-Joyce prgogram, the infinite time limit supposedly provides the Harder-Narasimhan decomposition. In general one cannot expect the special Lagrangian currents to be smooth, so this raises the question how to make sense of singular Lagrangians as representatives of $D^{b}Fuk(X)$ classes, or whether we should use some weaker equivalence class. Another interesting open problem is whether the limiting current is unique. In order to run the Thomas-Yau argument, one presumably also needs Floer theory for singular Lagrangians.

Joyce [41] already observed that it is not obvious how singular Lagrangians can carry brane structures, and it is logically possible for some Floer theoretic information to be lost in the infinite time limit. The suggestion is that hopefully the Lagrangian $L_{t}$ at large but finite time $t\gg 1$, can serve as a substitute for the infinite time limit, which presumably has better smoothness properties [41]. There is however no known justification (and probably false) that the surgeries terminate after some finite time, and $L_{t}$ decomposes into the union of several Lagrangian objects, in order to provide a Harder-Narasimhan decomposition.

We think Floer theory for singular Lagrangians is one of the foundational open questions necessary for an adequate solution of the Thomas-Yau conjecture. See section 5.4 for further discussions.