One start by calculating:

(3.2)

$$\begin{split}\Delta_{\varphi}(e^{-\alpha(F+\lambda\varphi)}(n+&\Delta\varphi))=\Delta_{\varphi}(e^{-\alpha(F+\lambda\varphi)})(n+\Delta\varphi)+e^{-\alpha(F+\lambda\varphi)}\Delta_{\varphi}(n+\Delta\varphi)\\ &+e^{-\alpha(F+\lambda\varphi)}(-\alpha)\frac{(F_{i}+\lambda\varphi_{i})(\Delta\varphi)_{\bar{i}}+(F_{\bar{i}}+\lambda\varphi_{\bar{i}})(\Delta\varphi)_{i}}{1+\varphi_{i\bar{i}}}.\end{split}$$

If we choose $\lambda>2\sup Ric$, then

(3.3)

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

For the term $\Delta_{\varphi}(n+\Delta\varphi)$, we choose a normal coordinate (c.f. equation (2.1)) and then follow Yau’s calculation[35]. First, note that

(3.4)

$$\Delta_{\varphi}(n+\Delta\varphi)=\frac{1}{1+\varphi_{k\bar{k}}}\bigg(g^{i\bar{j}}\varphi_{i\bar{j}}\bigg)_{k\bar{k}}=\frac{R_{i\bar{i}k\bar{k}}\varphi_{i\bar{i}}}{1+\varphi_{k\bar{k}}}+\frac{\varphi_{k\bar{k}i\bar{i}}}{1+\varphi_{k\bar{k}}}.$$

We wish to represent the $4$-th derivative of $\varphi$ in terms of $F$. For this we take equation (1.1) and differentiate it twice in $z_{i}$, $z_{\bar{i}}$ and then sum over $i=1,2\cdots n.\;$ We obtain:

(3.5)

$$\frac{\varphi_{k\bar{k}i\bar{i}}}{1+\varphi_{k\bar{k}}}-\frac{R_{k\bar{k}i\bar{i}}}{1+\varphi_{k\bar{k}}}-\frac{|\varphi_{k\bar{\beta}i}|^{2}}{(1+\varphi_{k\bar{k}})(1+\varphi_{\beta\bar{\beta}})}=F_{i\bar{i}}-R_{i\bar{i}}.$$

Hence

(3.6)

$$\begin{split}\Delta_{\varphi}(n&+\Delta\varphi)=\frac{R_{k\bar{k}i\bar{i}}(1+\varphi_{k\bar{k}})}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{p\bar{q}i}|^{2}}{(1+\varphi_{p\bar{p}})(1+\varphi_{q\bar{q}})}+\Delta F-R\\ &\geq-C_{3.1}(n+\Delta\varphi)\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{p\bar{q}i}|^{2}}{(1+\varphi_{p\bar{p}})(1+\varphi_{q\bar{q}})}+\Delta F-R.\end{split}$$

Here $C_{3.1}$ depends only on curvature bound of $g$ and $R$ is the scalar curvature of the background metric $g$. Plug in to equation (3.2) and we get

(3.7)

$$\begin{split}\Delta_{\varphi}&(e^{-\alpha(F+\lambda\varphi)}(n+\Delta\varphi))\geq e^{-\alpha(F+\lambda\varphi)}(\frac{\lambda\alpha}{2}-C_{3.1})(n+\Delta\varphi)\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}\\ &+\alpha e^{-\alpha(F+\lambda\varphi)}(\underline{R}-\lambda n)(n+\Delta\varphi)+e^{-\alpha(F+\lambda\varphi)}(\Delta F-R).\end{split}$$

Here we already drop the term:

$$\begin{split}&\frac{\alpha^{2}|F_{i}+\lambda\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}(n+\Delta\varphi)+(-\alpha)\frac{(F_{i}+\lambda\varphi_{i})(\Delta\varphi)_{\bar{i}}+(F_{\bar{i}}+\lambda\varphi_{\bar{i}})(\Delta\varphi)_{i}}{1+\varphi_{i\bar{i}}}+\frac{|\varphi_{p\bar{q}i}|^{2}}{(1+\varphi_{p\bar{p}})(1+\varphi_{q\bar{q}})}\\ \quad\geq&\frac{\alpha^{2}|F_{i}+\lambda\varphi_{i}|^{2}}{1+\varphi_{i\bar{i}}}(n+\Delta\varphi)-2\alpha Re\bigg(\frac{(F_{i}+\lambda\varphi_{i})(\Delta\varphi)_{\bar{i}}}{1+\varphi_{i\bar{i}}}\bigg)+\frac{|(\Delta\varphi)_{i}|^{2}}{(n+\Delta\varphi)(1+\varphi_{i\bar{i}})}\\ =&\frac{n+\Delta\varphi}{1+\varphi_{i\bar{i}}}|\alpha(F_{i}+\lambda\varphi_{i})-\frac{(\Delta\varphi)_{i}}{n+\Delta\varphi}|^{2}\geq 0.\end{split}$$

From the first line to second line in the above, we observed that

$$\frac{|(\Delta\varphi)_{i}|^{2}}{1+\varphi_{i\bar{i}}}=\frac{|\sum_{p}\varphi_{p\bar{p}i}|^{2}}{1+\varphi_{i\bar{i}}}\leq\frac{|\varphi_{p\bar{p}i}|^{2}(n+\Delta\varphi)}{(1+\varphi_{i\bar{i}})(1+\varphi_{p\bar{p}})}\\ \leq\frac{|\varphi_{p\bar{q}i}|^{2}(n+\Delta\varphi)}{(1+\varphi_{p\bar{p}})(1+\varphi_{q\bar{q}})}.$$

Set

$$u=e^{-\alpha(F+\lambda\varphi)}(n+\Delta\varphi)$$

and note that

$$(n+\Delta\varphi)\sum_{i}\frac{1}{1+\varphi_{i\bar{i}}}\geq e^{-\frac{F}{n-1}}(n+\Delta\varphi)^{1+\frac{1}{n-1}},$$

then we know from equation (3.7):

(3.8)

$$\begin{split}\Delta_{\varphi}&u\geq e^{-(\alpha+\frac{1}{n-1})F-\alpha\lambda\varphi}(\frac{\lambda\alpha}{2}-C_{3.1})(n+\Delta\varphi)^{1+\frac{1}{n-1}}-\alpha e^{-\alpha(F+\lambda\varphi)}(\lambda n-\underline{R})(n+\Delta\varphi)\\ &\qquad\qquad+e^{-\alpha(F+\lambda\varphi)}(\Delta F-R).\end{split}$$

We use the following equality, which holds for any $p\geq 0$:

$$\begin{array}[]{lcl}{1\over{2p+1}}\Delta_{\varphi}(u^{2p+1})&=&2pu^{2p-1}|\nabla_{\varphi}u|_{\varphi}^{2}+u^{2p}\Delta_{\varphi}u\\ &=&2pu^{2p-2}e^{-\alpha(F+\lambda\varphi)}(n+\Delta\varphi)|\nabla_{\varphi}u|_{\varphi}^{2}+u^{2p}\Delta_{\varphi}u\\ &\geq&2pu^{2p-2}|\nabla u|^{2}e^{-\alpha(F+\lambda\varphi)}+u^{2p}\Delta_{\varphi}u.\end{array}$$

Integrate with respect to $dvol_{\varphi}=e^{F}dvol_{g}$ and plug inequality (3.8) to get:

(3.9)

$$\begin{split}&\int_{M}2pu^{2p-2}|\nabla u|^{2}e^{(1-\alpha)F-\alpha\lambda\varphi}dvol_{g}\\ &\qquad\qquad\qquad\qquad+\int_{M}e^{-(\alpha-\frac{n-2}{n-1})F-\alpha\lambda\varphi}(\frac{\lambda\alpha}{2}-C_{3.1})(n+\Delta\varphi)^{1+\frac{1}{n-1}}u^{2p}dvol_{g}\\ &\qquad\qquad+\int_{M}e^{(1-\alpha)F-\alpha\lambda\varphi}u^{2p}\Delta Fdvol_{g}\leq\int_{M}\alpha e^{(1-\alpha)F-\alpha\lambda\varphi}(\lambda n-\underline{R})(n+\Delta\varphi)u^{2p}dvol_{g}\\ &\qquad\qquad\qquad\qquad\qquad\qquad+\int_{M}e^{(1-\alpha)F-\lambda\alpha\varphi}Ru^{2p}dvol_{g}.\end{split}$$

We need to handle the term involving $\Delta F$, which is done by integrating by parts.

(3.10)

$$\begin{split}&\int_{M}e^{(1-\alpha)F-\alpha\lambda\varphi}u^{2p}\Delta Fdvol_{g}=\int_{M}(\alpha-1)e^{(1-\alpha)F-\alpha\lambda\varphi}u^{2p}|\nabla F|^{2}dvol_{g}\\ &+\int_{M}\alpha\lambda e^{(1-\alpha)F-\lambda\alpha\varphi}u^{2p}\nabla\varphi\cdot\nabla Fdvol_{g}-\int_{M}2pe^{(1-\alpha)F-\lambda\alpha\varphi}u^{2p-1}\nabla u\cdot\nabla Fdvol_{g}.\end{split}$$

Also we can estimate the last term of (3.10)

(3.11)

$$u^{2p-1}\nabla u\cdot\nabla F\leq\frac{1}{2}u^{2p-2}|\nabla u|^{2}+\frac{1}{2}u^{2p}|\nabla F|^{2}.$$

Then we estimate the second to last term of (3.10) and obtain:

(3.12)

$$\begin{split}\alpha&\lambda e^{(1-\alpha)F-\lambda\alpha\varphi}u^{2p}\nabla\varphi\cdot\nabla F\\ \leq&\frac{\alpha-1}{2}e^{(1-\alpha)F-\lambda\alpha\varphi}u^{2p}|\nabla F|^{2}+\frac{\alpha^{2}\lambda^{2}}{2(\alpha-1)}u^{2p}|\nabla\varphi|^{2}e^{(1-\alpha)F-\lambda\alpha\varphi}\\ \leq&\frac{\alpha-1}{2}e^{(1-\alpha)F-\lambda\alpha\varphi}u^{2p}|\nabla F|^{2}+C_{2.3}\frac{\alpha^{2}\lambda^{2}}{2(\alpha-1)}u^{2p}e^{(2-\alpha)F-\lambda\alpha\varphi}.\end{split}$$

When estimating $|\nabla\varphi|^{2}$ above, we used Theorem 2.2, and $C_{2.3}$ is the constant given by that theorem. Plug (3.11), (3.12) back into (3.10), we obtain

(3.13)

$$\begin{split}&\int_{M}e^{(1-\alpha)F-\alpha\lambda\varphi}u^{2p}\Delta Fdvol_{g}\geq\int_{M}(\frac{\alpha-1}{2}-p)e^{(1-\alpha)F-\alpha\lambda\varphi}u^{2p}|\nabla F|^{2}dvol_{g}\\ &-\int_{M}C_{2.3}\frac{\alpha^{2}\lambda^{2}}{2(\alpha-1)}e^{(2-\alpha)F-\lambda\alpha\varphi}u^{2p}dvol_{g}-\int_{M}pe^{(1-\alpha)F-\lambda\alpha\varphi}u^{2p-2}|\nabla u|^{2}dvol_{g}.\end{split}$$

Plug (3.13) back to (3.9), we see

(3.14)

$$\begin{split}&\int_{M}pe^{(1-\alpha)F-\lambda\alpha\varphi}u^{2p-2}|\nabla u|^{2}dvol_{g}+\int_{M}(\frac{\alpha-1}{2}-p)e^{(1-\alpha)F-\alpha\lambda\varphi}u^{2p}|\nabla F|^{2}dvol_{g}\\ &\qquad\qquad+\int_{M}e^{-(\alpha-\frac{n-2}{n-1})F-\alpha\lambda\varphi}(\frac{\lambda\alpha}{2}-C_{3.1})(n+\Delta\varphi)^{1+\frac{1}{n-1}}u^{2p}dvol_{g}\\ &\leq\int_{M}\alpha e^{(1-\alpha)F-\alpha\lambda\varphi}(\lambda n-\underline{R})(n+\Delta\varphi)u^{2p}dvol_{g}+C_{2.3}\frac{\alpha^{2}\lambda^{2}}{2(\alpha-1)}\int_{M}e^{(2-\alpha)F-\lambda\alpha\varphi}u^{2p}dvol_{g}\\ &\qquad\qquad+\int_{M}e^{(1-\alpha)F-\lambda\alpha\varphi}Ru^{2p}dvol_{g}.\end{split}$$

Now let $\alpha>2p+1$ and $\lambda\alpha\geq 2C_{3.1}+1$, note that $n+\Delta\varphi$ has positive lower bound, then we find from above:

(3.15)

$$\begin{split}&\int_{M}e^{-(\alpha-\frac{n-2}{n-1})F-\alpha\lambda\varphi}(n+\Delta\varphi)^{1+\frac{1}{n-1}}u^{2p}dvol_{g}\\ &\qquad\leq C_{3.2}\alpha\int_{M}e^{(1-\alpha)F-\alpha\lambda\varphi}(n+\Delta\varphi)u^{2p}dvol_{g}+C_{3.2}\frac{\alpha^{2}}{\alpha-1}\int_{M}e^{(2-\alpha)F-\lambda\alpha\varphi}u^{2p}dvol_{g}.\end{split}$$

Recall the definition of $u$, this means for any $p\geq 0$, $\alpha\geq 2p+2$:

(3.16)

$$\begin{split}&\int_{M}\exp(-(2p+1)\alpha F+\frac{n-2}{n-1}F-\lambda\alpha(2p+1)\varphi)(n+\Delta\varphi)^{2p+1+\frac{1}{n-1}}dvol_{g}\\ &\leq C_{3.2}\alpha\int_{M}\exp(-(2p+1)\alpha F+F-(2p+1)\alpha\lambda\varphi)(n+\Delta\varphi)^{2p+1}dvol_{g}\\ &+C_{3.2}\frac{\alpha^{2}}{\alpha-1}\int_{M}\exp(-(2p+1)\alpha F+2F-(2p+1)\lambda\varphi)(n+\Delta\varphi)^{2p}dvol_{g}.\end{split}$$

Hence for some constant $C_{3.3}$ which depends on $||\varphi||_{0}$, $\alpha$, and $p$, we get:

(3.17)

$$\begin{split}&\int_{M}\exp(-(2p+1)\alpha F+\frac{n-2}{n-1}F)(n+\Delta\varphi)^{2p+1+\frac{1}{n-1}}dvol_{g}\\ &\leq C_{3.3}\bigg(\int_{M}\exp(-(2p+1)\alpha F+F)(n+\Delta\varphi)^{2p+1}dvol_{g}\\ &+\int_{M}\exp(-(2p+1)\alpha F+2F)(n+\Delta\varphi)^{2p}dvol_{g}\bigg).\end{split}$$

Start from $p=0$, and take $\alpha=2$, one obtains from (3.17) that:

(3.18)

$$\begin{split}\int_{M}e^{-\frac{n}{n-1}F}&(n+\Delta\varphi)^{\frac{n}{n-1}}dvol_{g}\leq C_{3.3}\big(\int_{M}e^{-F}(n+\Delta\varphi)dvol_{g}+\int_{M}dvol_{g}\big)\\ &\leq C_{3.3}\big(n||e^{-F}||_{0}vol(M)+vol(M)\big).\end{split}$$

Since we obtained in Proposition 2.1 a bound for $e^{-F}$ depending only on $||\varphi||_{0}$ and curvature bound of $g$. Hence we get a bound for $\int_{M}e^{-\frac{n}{n-1}F}(n+\Delta\varphi)^{\frac{n}{n-1}}dvol_{g}$.

We now claim that there exists a sequence of pair of positive numbers $(p_{k},\gamma_{k})$ where $p_{k}\rightarrow\infty$ such that

$$\displaystyle\int_{M}\;e^{-\gamma_{k}F}(n+\Delta\varphi)^{2p_{k}+1}\;dvol_{g}<\infty$$

for all $k=1,2\cdots.$ Now we explain how we choose this sequence of pairs of positive numbers successively: In general, suppose we already choose $(p_{k},\gamma_{k})$ such that the preceding inequality holds. Choose $\alpha_{k+1}$ sufficiently large such that

$$\alpha_{k+1}\geq 2p_{k}+2,\qquad{\rm and}\qquad-(2p_{k}+1)\alpha_{k+1}+2\leq-\gamma_{k}.$$

Set $\alpha=\alpha_{k+1}$, $p=p_{k}$ in (3.17), we obtain

(3.19)

$$\begin{split}&\int_{M}\exp(-(2p_{k}+1)\alpha_{k+1}F+\frac{n-2}{n-1}F)(n+\Delta\varphi)^{2p_{k}+1+\frac{1}{n-1}}dvol_{g}\\ &\leq C_{3.31}\bigg(\int_{M}e^{-\gamma_{k}F}(n+\Delta\varphi)^{2p_{k}+1}dvol_{g}+\int_{M}e^{-\gamma_{k}F}(n+\Delta\varphi)^{2p_{k}}dvol_{g}\bigg)\\ &\leq C_{3.32}\int_{M}e^{-\gamma_{k}F}(n+\Delta\varphi)^{2p_{k}+1}dvol_{g}.\end{split}$$

In the second inequality, we used again the fact that $e^{-F}$ is bounded in terms of $||\varphi||_{0}$ and $g$. In the last inequality above, we noticed the fact that $n+\Delta\varphi\geq e^{\frac{F}{n}}$, and $e^{F}$ is bounded from below. Set

$$\gamma_{k+1}=(2p_{k}+1)\alpha_{k+1}-\frac{n-2}{n-1}\qquad{\rm and}\qquad p_{k+1}=p_{k}+\frac{1}{2(n-1)}.$$

Then

$$\int_{M}e^{-\gamma_{k+1}F}(n+\Delta\varphi)^{2p_{k+1}+1}dvol_{g}\leq C\int_{M}e^{-\gamma_{k}F}(n+\Delta\varphi)^{2p_{k}+1}.$$

where the constant depends on $||\varphi||_{0}$ and the background metric $g$. Our claim is then verified.

By induction, we then get a bound for $\int_{M}e^{-\gamma_{p}F}(n+\Delta\varphi)^{p}dvol_{g}$ for any $p>0$ and some constant $\gamma_{p}>0$. Here $\gamma_{p}$ grows like $p^{2}$ as $p\rightarrow\infty$. ∎