In section 5.3 we discussed that the Solomon functional should be non-increasing under Joyce’s LMCF. Suppose the flow extends weakly to Lagrangians in $\mathcal{L}$. The flow starting from a minimizer $L$ must have constant $\mathcal{S}(L_{t})$, but the evolution (57) would then force $\theta=\hat{\theta}$, namely $L$ is a special Lagrangian, and the flow is in fact constant.

If the Lagrangian angle of the minimizer satisfies $-\pi/2+\epsilon<\inf_{L}\theta\leq\sup_{L}\theta<\pi/2-\epsilon$, then we have a more elliptic argument. Given any compactly supported global $C^{\infty}$ Hamiltonian function $H$ on $X$, we can associate a 1-parameter family of symplectomorphisms $\phi_{t}$ by exponentiating the Hamiltonian vector field. Since $d\phi_{t}$ only moves the tangent planes by $O(|t|)$ for small $|t|\ll 1$, the Lagrangian angle of $\phi_{t}(L)$ is still within $(-\pi/2+\epsilon,\pi/2-\epsilon)$, namely the quantitatively almost calibrated condition is preserved.

Under global Hamiltonian deformations, the first variation of the Solomon functional is

$$\delta S(H)=\frac{d}{dt}\mathcal{S}(\phi_{t}(L))|_{t=0}=\int_{L}H\text{Im}(e^{-i\hat{\theta}}\Omega).$$

We need another ingredient which is expected to hold once the Floer theory is sufficiently developed in the weak regularity setting:

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  The class of unobstructed exact Lagrangian objects is preserved by Hamiltonian isotopies. As such $\phi_{t}(L)$ should remain inside the class $\mathcal{L}$.

These would imply that the minimizer $L$ satisfies

$$\int_{L}H\text{Im}(e^{-i\hat{\theta}}\Omega)=0.$$

for any compactly supported $C^{\infty}$ function on $X$. This means $\text{Im}(e^{-i\hat{\theta}}\Omega)=0$ as currents, which is equivalent to $\theta=\hat{\theta}$ under the almost calibrated setting.