This step is relatively easy. Let $p\in M$, we may choose a local normal coordinate in a neighborhood of $p$, such that

(2.1)

$$g_{i\bar{j}}(p)=\delta_{ij},\;\;\nabla g_{i\bar{j}}(p)=0,\;\;{\rm and}\;\;\varphi_{i\bar{j}}(p)=\varphi_{i\bar{i}}(p)\delta_{ij}.$$

In this paper, we will always work under this coordinate unless specified otherwise. Choose the constant $C_{2.1}$ to be $C_{2.1}=2\displaystyle\max_{M}R_{i\bar{i}}+\frac{2|\underline{R}|}{n}+1$. Under this coordinate, we can calculate:

(2.2)

$$\begin{split}\Delta_{\varphi}(F+C_{2.1}\varphi)&=-\underline{R}+\frac{R_{i\bar{i}}}{1+\varphi_{i\bar{i}}}+C_{2.1}\frac{\varphi_{i\bar{i}}}{1+\varphi_{i\bar{i}}}\\ &=-\underline{R}+C_{2.1}n-\frac{C_{2.1}-R_{i\bar{i}}}{1+\varphi_{i\bar{i}}}\leq-\underline{R}+C_{2.1}n-\frac{nC_{2.1}}{2}e^{-\frac{F}{n}}\\ &\leq 2C_{2.1}n-\frac{nC_{2.1}}{2}e^{-\frac{F}{n}}.\end{split}$$

In the second line above, we used the arithemetic-geometric inequality:

$$\frac{1}{n}\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}\geq\Pi_{i}(1+\varphi_{i\bar{i}})^{-\frac{1}{n}}=e^{-\frac{F}{n}}.$$

Now let $p_{0}$ be such that the function $F+C_{2.1}\varphi$ achieves minimum at $p_{0}$, then from (2.2), we see

(2.3)

$$0\leq 2C_{2.1}n-\frac{nC_{2.1}}{2}e^{-\frac{F}{n}}(p_{0}).$$

This gives a lower bound for $F$, depending only on $C^{0}$ bound for $\varphi$.

∎