If $\omega_{E}|_{\bar{S}}\neq 0$, and thus $\omega_{0}|_{S}\neq 0$, then there are two vectors $v_{1},v_{2}\in\bar{S}$ such that $\omega_{E}(v_{1},v_{2})>0$. By perturbing $v_{1}$ and $v_{2}$ a little bit if necessary, we have that $\tilde{\Sigma}=\mathfrak{q}(\{t_{1}v_{1}+t_{2}v_{2}|t_{i}\in\mathbb{R}\})$ is a closed 2-torus in $\tilde{S}$, i.e. a closed 2-parameters subgroup. Thus $\Sigma=\pi_{h}(\tilde{\Sigma})$ is a closed oriented surface in $S$, which satisfies

$$\int_{\Sigma}\omega_{0}\geq\frac{1}{\lambda(n)}\int_{\tilde{\Sigma}}\pi_{h}^{*}\omega_{0}\geq\frac{\omega_{E}(v_{1},v_{2})}{\lambda(n)\|v_{1}\wedge v_{2}\|_{h_{E}}}V_{E}>0,$$

where $V_{E}$ denotes the Euclidean area of the intersection of $\{t_{1}v_{1}+t_{2}v_{2}|t_{i}\in\mathbb{R}\}$ with the fundamental domain of the quotient map $\mathfrak{q}$. From the smooth convergence of $(M_{k},\tilde{\omega}_{k},\tilde{g}_{k})$,

$$\lim_{k\longrightarrow\infty}i^{-2}_{g_{k}}(p_{k})\int_{F_{r,k}(\Sigma)}\omega_{k}=\lim_{k\longrightarrow\infty}\int_{F_{r,k}(\Sigma)}\tilde{\omega}_{k}=\lim_{k\longrightarrow\infty}\int_{\Sigma}F_{r,k}^{*}\tilde{\omega}_{k}=\int_{\Sigma}\omega_{0},$$

for $r\gg 1$ such that $\Sigma\subset B_{g_{0}}(p_{0},r)$. Thus

$$0<\frac{1}{2}i^{2}_{g_{k}}(p_{k})\int_{\Sigma}\omega_{0}\leq\int_{F_{r,k}(\Sigma)}\omega_{k}\leq 2i^{2}_{g_{k}}(p_{k})\int_{\Sigma}\omega_{0}\leq 2k^{-2}\int_{\Sigma}\omega_{0}<1,$$

for $k\gg 1$. Since $[F_{r,k}(\Sigma)]\in H_{2}(M_{k},\mathbb{Z})$ and $[\omega_{k}]\in H^{2}(M_{k},\mathbb{Z})$, we obtain

$$\int_{F_{r,k}(\Sigma)}\omega_{k}\in\mathbb{Z},$$

which is a contradiction. Hence $\omega_{E}|_{\bar{S}}\equiv 0$ and $\omega_{0}|_{S}\equiv 0$, which implies that $S$ is a lagrangian submanifold $(X,\omega_{0})$ by combining Lemma 3.1.

Since $\bar{S}$ is a lagrangian linear subspace of $(\mathbb{C}^{n},\omega_{E})$, there is a $\theta_{0}\in\mathbb{R}$ such that ${\rm Im}e^{\sqrt{-1}\theta_{0}}\Omega_{E}|_{\bar{S}}=0$. This implies that $\bar{S}$ is a special lagrangian linear subspace of phase $\theta_{0}$ in $(\mathbb{C}^{n},\omega_{E},\Omega_{E})$. Thus $S$ is a special lagrangian submanifold of phase $\theta_{0}$ in $(X,\omega_{0},\Omega_{0})$, i.e.

$$\omega_{0}|_{S}=0,\ \ \ {\rm Im}e^{\sqrt{-1}\theta_{0}}\Omega_{0}|_{S}=0.$$

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