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. ∎