(Heuristic) In Joyce’s conjectural program, the Harder-Narasimhan decomposition is constructed by running the LMCF $(L_{t})$ starting from the unobstructed Lagrangian $L$, and take the infinite time limit (6) to obtain the limiting special Lagrangians $L_{1},\ldots,L_{N}$ with angles $\hat{\theta}_{1}>\hat{\theta}_{2}>\ldots>\hat{\theta}_{N}$, assuming $L_{1},\ldots L_{N}$ have enough regularity to be admitted as objects of $D^{b}Fuk(X)$. It is expected that $L_{1},\ldots L_{N}$ generate $L$ in $D^{b}Fuk(X)$ via (4).

A basic feature of LMCF in Calabi-Yau manifolds is that the Lagrangian angle satisfies a heat equation (cf. section 4.1), so $\sup_{L_{t}}\theta$ is nonincreasing in time (resp. $\inf_{L_{t}}\theta$ is nondecreasing). Comparing the initial time with the infinite time limit, this suggests $\sup\theta_{L}\geq\hat{\theta}_{1}$ and $\inf\theta_{L}\leq\hat{\theta}_{N}$.

Morever, if the ambient metric is Calabi-Yau, then LMCF is a special case of mean curvature flow, so the volume functional decreases in time. This monotonicity is not affected by the surgeries in Joyce’s LMCF. Thus

$$\text{Vol}(L)\geq\sum_{1}^{N}\text{Vol}(L_{i})=\sum_{1}^{N}|Z(L_{i})|=\sum_{1}^{N}|\int_{L_{i}}\Omega|.$$

Under the Calabi-Yau metric, the volume of the Lagrangian $L$ agrees with the $J$-volume:

$$\text{Vol}(L)=\int_{L}e^{-i\theta}\Omega=\int_{L}|\Omega|.$$

so (36) follows. ∎