(Floer theoretic obstruction) Assume the automatic transversality and the positivity condition hold for the bordism current $\mathcal{C}$. Assume the destabilizing condition

$$\hat{\theta}_{1}=\arg\int_{L_{1}}\Omega>\hat{\theta}_{2}=\arg\int_{L_{2}}\Omega.$$

Then the Lagrangian phase angle of $L$ has a lower bound on its oscillation:

$$\sup_{L}\theta_{L}\geq\hat{\theta}_{1},\quad\inf_{L}\theta_{L}\leq\hat{\theta}_{2},$$

(31)

and morever the J-volume of $L$ (cf. section 2.9) has a nontrivial lower bound

$$\text{Vol}_{J}(L)=\int_{L}e^{-i\theta}\Omega\geq|\int_{L_{1}}\Omega|+|\int_{L_{2}}\Omega|.$$

(32)