From $e^{i\theta}dvol_{L}=\Omega|_{L}=\text{Re}\Omega|_{L}+\sqrt{-1}\text{Im}\Omega|_{L},$ we see $\text{Re }\Omega$ is an orientation form on $L$, and we arrive at

$$\text{Vol}(L)=\int_{L}\frac{1}{\cos\theta}\text{Re }\Omega\leq\frac{1}{\sin\epsilon}\int_{L}\text{Re }\Omega$$

as required. ∎