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 $\varphi$.

∎

The argument uses maximum principle again. This time we will calculate $\Delta_{\varphi}(e^{F-\lambda\varphi}(K+|\nabla\varphi|^{2}))$ where $\lambda,K>0$ are constants to be determined later. We choose a normal coordinate (equation (2.1)) and do the following calculations. We have

(2.4)		$\displaystyle\qquad\Delta_{\varphi}(e^{F-\lambda\varphi}(K+|\nabla\varphi|^{2}))$	$\displaystyle=$	$\displaystyle\Delta_{\varphi}(e^{F-\lambda\varphi})(K+|\nabla\varphi|^{2})+e^{F-\lambda\varphi}\Delta_{\varphi}(K+|\nabla\varphi|^{2})$
				$\displaystyle+e^{F-\lambda\varphi}\cdot\frac{(F_{i}-\lambda\varphi_{i})(|\nabla\varphi|^{2})_{\bar{i}}+(F_{\bar{i}}-\lambda\varphi_{\bar{i}})(|\nabla\varphi|^{2})_{i}}{1+\varphi_{i\bar{i}}}.$

We can first calculate:

(2.5)

$$\begin{split}\Delta_{\varphi}(e^{F-\lambda\varphi})&=e^{F-C\varphi}\frac{|F_{i}-\lambda\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+e^{F-\lambda\varphi}(\Delta_{\varphi}F-\frac{\lambda\varphi_{i\bar{i}}}{1+\varphi_{i\bar{i}}})\\ &=e^{F-\lambda\varphi}\frac{|F_{i}-\lambda\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+e^{F-\lambda\varphi}(-\underline{R}-\lambda n+\frac{\lambda+R_{i\bar{i}}}{1+\varphi_{i\bar{i}}}).\end{split}$$

By differentiating equation (1.1) in $z_{\alpha}$ direction, we obtain

(2.6)

$$\displaystyle\sum_{i}\frac{\varphi_{i\bar{i}\alpha}}{1+\varphi_{i\bar{i}}}=F_{\alpha}\;\;{\rm and}\;\;\displaystyle\sum_{i}\frac{\varphi_{i\bar{i}\bar{\alpha}}}{1+\varphi_{i\bar{i}}}=F_{\bar{\alpha}}.$$

Then we calculate

(2.7)

$$\begin{split}\Delta_{\varphi}(|\nabla\varphi|^{2})&=\frac{R_{\alpha\bar{\beta}i\bar{i}}\varphi_{\alpha}\varphi_{\bar{\beta}}}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{\varphi^{2}_{i\bar{i}}}{1+\varphi_{i\bar{i}}}+\frac{\varphi_{\alpha}\varphi_{\bar{\alpha}i\bar{i}}+\varphi_{\bar{\alpha}}\varphi_{\alpha i\bar{i}}}{1+\varphi_{i\bar{i}}}\\ &\geq-\frac{C_{2.21}|\nabla\varphi|^{2}}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{\varphi^{2}_{i\bar{i}}}{1+\varphi_{i\bar{i}}}+\varphi_{\alpha}F_{\bar{\alpha}}+\varphi_{\bar{\alpha}}F_{\alpha}\\ &=-\frac{C_{2.21}|\nabla\varphi|^{2}}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{\varphi^{2}_{i\bar{i}}}{1+\varphi_{i\bar{i}}}+\varphi_{\alpha}(F_{\bar{\alpha}}-\lambda\varphi_{\bar{\alpha}})+\varphi_{\bar{\alpha}}(F_{\alpha}-\lambda\varphi_{\alpha})+2\lambda|\varphi_{\alpha}|^{2}\\ &\geq-\frac{C_{2.21}|\nabla\varphi|^{2}}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{\varphi^{2}_{i\bar{i}}}{1+\varphi_{i\bar{i}}}-\varepsilon(n+\Delta\varphi)-\frac{|F_{\alpha}-\lambda\varphi_{\alpha}|^{2}|\varphi_{\alpha}|^{2}}{\varepsilon(1+\varphi_{\alpha\bar{\alpha}})}+2\lambda|\varphi_{\alpha}|^{2}.\end{split}$$

Here $C_{2.21}$ depends only on lower bound of bisectional curvature of $g$.

For the last term in (2.4), we estimate in the following way:

(2.8)				$\displaystyle\frac{|(F_{\bar{i}}-\lambda\varphi_{\bar{i}})(|\nabla\varphi|^{2})_{i}|}{1+\varphi_{i\bar{i}}}$
			$\displaystyle=$	$\displaystyle\frac{|(F_{\bar{i}}-\lambda\varphi_{\bar{i}})(\varphi_{\alpha}\varphi_{\bar{\alpha}i}+\varphi_{\bar{\alpha}}\varphi_{\alpha i})|}{1+\varphi_{i\bar{i}}}$
			$\displaystyle\leq$	$\displaystyle\frac{|(F_{\bar{i}}-\lambda\varphi_{\bar{i}})\varphi_{i}\varphi_{i\bar{i}}|}{1+\varphi_{i\bar{i}}}+\frac{|(F_{\bar{i}}-\lambda\varphi_{\bar{i}})\varphi_{\bar{\alpha}}\varphi_{\alpha i}|}{1+\varphi_{i\bar{i}}}$
			$\displaystyle\leq$	$\displaystyle\frac{|F_{i}-\lambda\varphi_{i}|^{2}|\varphi_{i}|^{2}}{2\varepsilon(1+\varphi_{i\bar{i}})}+\frac{|F_{i}-\lambda\varphi_{i}|^{2}|\varphi_{\alpha}|^{2}}{2\varepsilon(1+\varphi_{i\bar{i}})}+\frac{\varepsilon\varphi_{i\bar{i}}^{2}}{2(1+\varphi_{i\bar{i}})}+\frac{\varepsilon|\varphi_{i\alpha}|^{2}}{2(1+\varphi_{i\bar{i}})}.$

The other conjugate term satisfies the same estimate as above. Combining above calculations, we obtain:

(2.9)				$\displaystyle\frac{\Delta_{\varphi}(e^{F-\lambda\varphi}(K+|\nabla\varphi|^{2}))}{e^{F-\lambda\varphi}}$
			$\displaystyle\geq$	$\displaystyle\big(K+|\nabla\varphi|^{2}-3\varepsilon^{-1}|\nabla\varphi|^{2}\big)\frac{|F_{i}-\lambda\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}$
				$\displaystyle\qquad\qquad+\frac{(\lambda+R_{i\bar{i}})(K+|\nabla\varphi|^{2})-C_{2.21}|\nabla\varphi|^{2}}{1+\varphi_{i\bar{i}}}+\frac{(1-\varepsilon)\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}-\varepsilon(n+\Delta\varphi)$
				$\displaystyle\qquad\qquad\qquad\qquad+\frac{|\varphi_{i\alpha}|^{2}(1-\varepsilon)}{1+\varphi_{i\bar{i}}}+(-\underline{R}-\lambda n)(K+|\nabla\varphi|^{2}).$

Now it’s time to choose the constants $\varepsilon$, $\lambda$, and $K$ appearing above.

First we choose $\varepsilon=\frac{1}{4}$. With this choice, we have

(2.10)

$$\begin{split}\sum_{i}\frac{(1-\varepsilon)\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}-\varepsilon(n+\Delta\varphi)&=\frac{3}{4}(n+\Delta\varphi)-\frac{3n}{2}+\frac{3}{4}\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}-\frac{1}{4}(1+\varphi_{i\bar{i}})\\ &\geq\frac{1}{2}(n+\Delta\varphi)-\frac{3n}{2}.\end{split}$$

Then we choose $\lambda$ so large that $\lambda+R_{i\bar{i}}>1$. Finally, we choose $K$ so large that

(2.11)			$\displaystyle K>C_{2.21}\max_{M}|\nabla\varphi|^{2}$
(2.12)			$\displaystyle K>3\varepsilon^{-1}\max_{M}|\nabla\varphi|^{2}=12\max_{M}|\nabla\varphi|^{2}.$

With above choices for $\lambda$ and $K$, we have

(2.13)

$$(\lambda+R_{i\bar{i}})(K+|\nabla\varphi|^{2})-C_{2.21}|\nabla\varphi|^{2}\geq K-C_{2.21}|\nabla\varphi|^{2}>0.$$

and also

(2.14)

$$K-3\varepsilon^{-1}|\nabla\varphi|^{2}>0.$$

Hence we conclude from (2.9) that

(2.15)

$$\Delta_{\varphi}(e^{F-\lambda\varphi}(K+|\nabla\varphi|^{2}))\geq e^{F-\lambda\varphi}\big(-(|\underline{R}|+\lambda n)(K+\max_{M}|\nabla\varphi|^{2})-C_{2.22}+\frac{1}{4}(n+\Delta\varphi)\big).$$

Denote $v=e^{F-C\varphi}(K+|\nabla\varphi|^{2})$, it is enough to show $v$ has an upper bound. we see from (2.15) that there exists constants $C_{2.23}>0$, $C_{2.24}>0$, possibly depending on $\displaystyle\max_{M}|\nabla\varphi|^{2}$, such that

(2.16)

$$\Delta_{\varphi}(v)\geq v(-C_{2.23}+\frac{1}{C_{2.24}}(n+\Delta\varphi)).$$

Here we notice that $n+\Delta\varphi\geq ne^{\frac{F}{n}}$. Hence we obtain from (2.16) that

(2.17)

$$\Delta_{\varphi}(v)\geq v(-C_{2.23}+\frac{1}{C_{2.24}}e^{\frac{F}{n}}).$$

Let the maximum of $v$ be achieved at point $p$, then we know $-C_{2.23}+\frac{e^{\frac{F}{n}(p)}}{C_{2.24}}\leq 0$. This gives an upper bound of $F$ at $p_{0}$, hence an upper bound for $v$, where this bound depends on $\displaystyle\max_{M}|\nabla\varphi|$. ∎

We will consider $\Delta_{\varphi}(e^{-(F+\lambda\varphi)+\frac{1}{2}\varphi^{2}}(|\nabla\varphi|^{2}+K))$. Here $\lambda>0$, $K>0$ are constants to be determined below. Then we have

(2.18)				$\displaystyle\Delta_{\varphi}(e^{-(F+\lambda\varphi)+\frac{1}{2}\varphi^{2}}(|\nabla\varphi|^{2}+K))$
			$\displaystyle=$	$\displaystyle\Delta_{\varphi}(e^{-(F+\lambda\varphi)+\frac{1}{2}\varphi^{2}})(|\nabla\varphi|^{2}+K)+e^{-(F+\lambda\varphi)+\frac{1}{2}\varphi^{2}}\Delta_{\varphi}(|\nabla\varphi|^{2})$
				$\displaystyle\qquad\qquad+\frac{2e^{-(F+\lambda\varphi)+\frac{1}{2}\varphi^{2}}}{1+\varphi_{i\bar{i}}}Re\big((-F_{i}-\lambda\varphi_{i}+\varphi\varphi_{i})(|\nabla\varphi|^{2})_{\bar{i}}\big).$

For simplicity of notation, set

$$A(F,\varphi)=-(F+\lambda\varphi)+\frac{1}{2}\varphi^{2}.$$

Similar as before, we may calculate:

(2.19)				$\displaystyle\Delta_{\varphi}(e^{A(F,\varphi)})$
			$\displaystyle=$	$\displaystyle e^{A}\frac{|-F_{i}-\lambda\varphi_{i}+\varphi\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+e^{A}\big(-\Delta_{\varphi}(F+\lambda\varphi)+\varphi\Delta_{\varphi}\varphi\big)+e^{A}\frac{|\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}$
			$\displaystyle=$	$\displaystyle e^{A}\frac{|-F_{i}-\lambda\varphi_{i}+\varphi\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+e^{A}\bigg(\underline{R}-\lambda n+n\varphi+\sum_{i}\frac{\lambda-R_{i\bar{i}}-\varphi}{1+\varphi_{i\bar{i}}}\bigg)+\frac{e^{A}|\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}.$

Recall the calculation in (2.7):

(2.20)

$$\begin{split}\Delta_{\varphi}(|\nabla\varphi|^{2})&=\frac{R_{i\bar{i}\alpha\bar{\beta}}\varphi_{\alpha}\varphi_{\bar{\beta}}}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}+F_{\bar{\alpha}}\varphi_{\alpha}+F_{\alpha}\varphi_{\bar{\alpha}}\\ &\geq-C_{2.21}|\nabla\varphi|^{2}\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}\\ &+(-2\lambda+2\varphi)|\nabla\varphi|^{2}+2Re\big((F_{\alpha}+\lambda\varphi_{\alpha}-\varphi\varphi_{\alpha})\varphi_{\bar{\alpha}}\big).\end{split}$$

Again $C_{2.21}$ depends only on curvature bound of $g$. Also

$$(|\nabla\varphi|^{2})_{\bar{i}}=\varphi_{\alpha}\varphi_{\bar{\alpha}\bar{i}}+\varphi_{\bar{i}}\varphi_{i\bar{i}},\,\,(|\nabla\varphi|^{2})_{i}=\varphi_{\bar{\alpha}}\varphi_{\alpha i}+\varphi_{i}\varphi_{i\bar{i}}.$$

Hence if we plug in (2.19) and (2.20) back to (2.18), we obtain:

(2.21)

$$\begin{split}&\quad\Delta_{\varphi}(e^{A}(|\nabla\varphi|^{2}+K))e^{-A}\\ &\geq|\nabla_{\varphi}(F+\lambda\varphi)-\varphi\nabla_{\varphi}\varphi|^{2}(|\nabla\varphi|^{2}+K)+|\nabla_{\varphi}\varphi|^{2}(|\nabla\varphi|^{2}+K)\\ &+\big(\underline{R}-\lambda n+n\varphi+\sum_{i}\frac{\lambda-R_{i\bar{i}}-\varphi}{1+\varphi_{i\bar{i}}}\big)(|\nabla\varphi|^{2}+K)+\frac{-C_{2.21}|\nabla\varphi|^{2}+|\varphi_{i\alpha}|^{2}+\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}\\ &+(-2\lambda+2\varphi)|\nabla\varphi|^{2}+2Re\big((F_{\alpha}+\lambda\varphi_{\alpha}-\varphi\varphi_{\alpha})\varphi_{\bar{\alpha}}\big)\\ &\qquad\qquad\qquad\qquad\qquad+\frac{2Re\big((-F_{i}-\lambda\varphi_{i}+\varphi\varphi_{i})(\varphi_{\alpha}\varphi_{\bar{\alpha}\bar{i}}+\varphi_{\bar{i}}\varphi_{i\bar{i}})\big)}{1+\varphi_{i\bar{i}}}.\end{split}$$

We notice the following complete square in the above sum:

(2.22)

$$\begin{split}&\frac{1}{1+\varphi_{i\bar{i}}}|\varphi_{i\alpha}-\big(F_{i}+\lambda\varphi_{i}-\varphi\varphi_{i}\big)\varphi_{\alpha}|^{2}\\ &=\frac{|\varphi_{i\alpha}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{2Re\big((-F_{i}-\lambda\varphi_{i}+\varphi\varphi_{i})\varphi_{\alpha}\varphi_{\bar{\alpha}\bar{i}}\big)}{1+\varphi_{i\bar{i}}}+\frac{|-F_{i}-\lambda\varphi_{i}+\varphi\varphi_{i}|^{2}|\nabla\varphi|^{2}}{1+\varphi_{i\bar{i}}}.\end{split}$$

We will drop this complete square in the following. Next we observe a crucial cancellation, which is the key point of this argument. We look at the last two terms in (2.21) and observe:

(2.23)

$$(F_{\alpha}+\lambda\varphi_{\alpha}-\varphi\varphi_{\alpha})\varphi_{\bar{\alpha}}+\frac{(-F_{i}-\lambda\varphi_{i}+\varphi\varphi_{i})\varphi_{\bar{i}}\varphi_{i\bar{i}}}{1+\varphi_{i\bar{i}}}=\frac{(F_{i}+\lambda\varphi_{i}-\varphi\varphi_{i})\varphi_{\bar{i}}}{1+\varphi_{i\bar{i}}}.$$

Hence we have

(2.24)

$$\begin{split}&\Delta_{\varphi}(e^{A}(|\nabla\varphi|^{2}+K))e^{-A}\geq K\frac{|-F_{i}-\lambda\varphi_{i}+\varphi\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{i}|^{2}(|\nabla\varphi|^{2}+K)}{1+\varphi_{i\bar{i}}}\\ &+\sum_{i}\frac{\lambda-R_{i\bar{i}}-\varphi}{1+\varphi_{i\bar{i}}}(|\nabla\varphi|^{2}+K)+\bigg(\underline{R}-\lambda n+n\varphi\bigg)(|\nabla\varphi|^{2}+K)\\ &-C_{2.21}|\nabla\varphi|^{2}\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}+\frac{\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}+(-2\lambda+2\varphi)|\nabla\varphi|^{2}+2Re\bigg(\frac{(F_{i}+\lambda\varphi_{i}-\varphi\varphi_{i})\varphi_{\bar{i}}}{1+\varphi_{i\bar{i}}}\bigg).\end{split}$$

Now we make the choices of $\lambda$, $K$. We choose $\lambda=10(\sup_{M}|R_{i\bar{i}}|+||\varphi||_{0}+C_{2.21}+1)$ and $K=10$. With this choice, we now estimate the terms in (2.24), with various constants $C_{i}$ which depends only on the curvature bound of $g$ and $||\varphi||_{0}$.

(2.25)

$$\big(\underline{R}-\lambda n+n\varphi\big)(|\nabla\varphi|^{2}+K)\geq-C_{2.31}(|\nabla\varphi|^{2}+1).$$

(2.26)

$$(-2\lambda+2\varphi)|\nabla\varphi|^{2}\geq-C_{2.32}|\nabla\varphi|^{2}.$$

(2.27)

$$\begin{split}\frac{|(F_{i}+\lambda\varphi_{i}-\varphi\varphi_{i})\varphi_{\bar{i}}|}{1+\varphi_{i\bar{i}}}&\leq\frac{1}{2}\frac{|F_{i}+\lambda\varphi_{i}-\varphi\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{1}{2}\frac{|\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}\\ &\leq\frac{1}{2}\frac{|F_{i}+\lambda\varphi_{i}-\varphi\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{1}{2}|\nabla\varphi|^{2}\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}.\end{split}$$

(2.28)

$$\sum_{i}\frac{\lambda-R_{i\bar{i}}-\varphi}{1+\varphi_{i\bar{i}}}(|\nabla\varphi|^{2}+K)-C_{2.21}|\nabla\varphi|^{2}\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}\geq 10|\nabla\varphi|^{2}\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}.$$

(2.29)

$$\frac{\varphi_{i\bar{i}}^{2}}{1+\varphi_{i\bar{i}}}\geq 0.$$

Combining all these estimates, we obtain from (2.24) that

(2.30)

$$\Delta_{\varphi}(e^{A}(|\nabla\varphi|^{2}+K))\geq e^{A}\big(\frac{|\varphi_{i}|^{2}|\nabla\varphi|^{2}}{1+\varphi_{i\bar{i}}}+9|\nabla\varphi|^{2}\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}-C_{2.33}(|\nabla\varphi|^{2}+1)\big).$$

Here $C_{2.33}$ depends only on curvature bound of $g$ and $||\varphi||_{0}$. Using Young’s inequality, we have,

$$\begin{split}|\nabla\varphi|^{\frac{2}{n}}e^{-\frac{F}{n}}&\leq\sum_{i}\frac{|\varphi_{i}|^{\frac{2}{n}}}{(1+\varphi_{i\bar{i}})^{\frac{1}{n}}}\cdot(1+\varphi_{i\bar{i}})^{\frac{1}{n}}e^{-\frac{F}{n}}\\ &\leq\frac{1}{n}\sum_{i}\frac{|\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{n-1}{n}\sum_{i}(1+\varphi_{i\bar{i}})^{\frac{1}{n-1}}e^{-\frac{F}{n-1}}\\ &\leq(n-1)\big(\frac{|\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+\frac{1}{n}\sum_{i}(1+\varphi_{i\bar{i}})^{\frac{1}{n-1}}e^{-\frac{F}{n-1}}\big)\\ &\leq(n-1)\big(\frac{|\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+(n+\Delta\varphi)^{\frac{1}{n-1}}e^{-\frac{F}{n-1}}\big)\\ &\leq(n-1)\big(\frac{|\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}+\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}\big).\end{split}$$

Thus,

$$\begin{split}\frac{|\varphi_{i}|^{2}|\nabla\varphi|^{2}}{1+\varphi_{i\bar{i}}}+|\nabla\varphi|^{2}\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}&\geq\frac{1}{n-1}|\nabla\varphi|^{2+\frac{2}{n}}e^{-\frac{F}{n}}.\end{split}$$

Hence we get from (2.30) that

(2.31)

$$\begin{split}\Delta_{\varphi}(e^{A}(|\nabla\varphi|^{2}+K))&\geq e^{-C\varphi+\frac{1}{2}\varphi^{2}}\big(e^{-(1+\frac{1}{n})F}|\nabla\varphi|^{2+\frac{2}{n}}-C_{2.33}e^{-F}|\nabla\varphi|^{2}-C_{2.33}e^{-F}\big)\\ &=e^{-C\varphi+\frac{1}{2}\varphi^{2}}\big((e^{-F}|\nabla\varphi|^{2})^{1+\frac{1}{n}}-C_{2.33}e^{-F}|\nabla\varphi|^{2}-C_{2.33}e^{-F}\big).\end{split}$$

Suppose that the function $e^{-(F+C\varphi)+\frac{1}{2}\varphi^{2}}(|\nabla\varphi|^{2}+K)$ achieves maximum at $p$. Then at point $p$, we have

(2.32)

$$0\geq(e^{-F}|\nabla\varphi|^{2})^{1+\frac{1}{n}}-C_{2.33}e^{-F}|\nabla\varphi|^{2}-C_{2.33}e^{-F}.$$

Recall Proposition 2.1 gives an estimate for $e^{-F}$ which depends only on $||\varphi||_{0}$ and the curvature bound of $g$. Therefore, we get a bound for $e^{-F}|\nabla\varphi|^{2}(p)$ with the same dependence. Hence we have a bound for $e^{-(F+\lambda\varphi)+\frac{1}{2}\varphi^{2}}(|\nabla\varphi|^{2}+K)(p)$, with the dependence as stated in the theorem.

But this function achieves maximum at $p$, so we are done. ∎