In this section (Proof of Theorem 5.2 and Lemma 5.3), we apply this estimate with $d=2n$ to the operator $\Delta_{\varphi}$. After rewriting $\Delta_{\varphi}$ in terms of real coefficients, one can find $D^{*}=\big(\det(g_{\varphi})_{i\bar{j}}\big)^{-\frac{1}{n}}=e^{-\frac{F}{n}}\big(\det g_{i\bar{j}}\big)^{-\frac{1}{n}}$.