ScalingStacks

Next, we show (2.11). Recall from (3.10) and (3.24) that on $X\backslash S$ we have

(3.29)

$$\textrm{tr}_{\omega_{X}}\tilde{\omega}_{t}\leq Ce^{C_{0}e^{B\sigma^{-\lambda}}}.$$

(3.30)

$$\textrm{tr}_{\tilde{\omega}_{t}}\omega_{X}\leq\frac{C}{t}e^{C_{0}e^{B\sigma^{-\lambda}}},$$

for uniform constants $B,C,C_{0}$. We apply the maximum principle to the quantity

$$K_{2}=e^{-Ae^{B\sigma^{-\lambda}}}\left(\mathcal{S}+C\frac{e^{3C_{0}e^{B\sigma^{-\lambda}}}}{t^{5/2}}\textrm{tr}_{\omega_{X}}\tilde{\omega}_{t}\right),$$

for suitable constants $A,C$, where the quantity $\mathcal{S}$ is the same quantity as in [Y1]:

$$\mathcal{S}=|\nabla\tilde{\omega}_{t}|^{2}_{\tilde{\omega}_{t}},$$

where $\nabla$ is the covariant derivative associated to the metric $\omega_{X}$. Using $\varphi_{t}$ we can write

$$\mathcal{S}=\tilde{g}_{t}^{i\overline{p}}\tilde{g}_{t}^{q\overline{j}}\tilde{g}_{t}^{k\overline{r}}\varphi_{i\overline{j}k}\varphi_{\overline{p}q\overline{r}},$$

where again lower indices are covariant derivatives with respect to $\omega_{X}$. We are going to show that $K_{2}\leq\frac{C}{t^{5/2}}$, and using (3.29) this implies that

(3.31)

$$\mathcal{S}\leq\frac{Ce^{Ae^{B\sigma^{-\lambda}}}}{t^{5/2}}.$$

We now use (3.28), which says that on $X_{y}$ we have

(3.32)

$$\textrm{tr}_{\omega_{y}}\tilde{\omega}_{y}\leq tCe^{C_{0}e^{B\sigma^{-\lambda}}},$$

At any given point of $X_{y}$ we can assume that $\omega_{X}$ is the identity and $\tilde{\omega}_{t}$ is diagonal with positive entries $\lambda_{i}$, $1\leq i\leq n$, so that the first $n-m$ directions are tangent to the fiber $X_{y}$. Then (3.32) gives that

(3.33)

$$\lambda_{i}\leq tCe^{C_{0}e^{B\sigma^{-\lambda}}},$$

for $1\leq i\leq n-m$. Then using (3.31) we see that

$$\sum_{i,j,k=1}^{n-m}\frac{1}{\lambda_{i}\lambda_{j}\lambda_{k}}|\varphi_{i\overline{j}k}|^{2}\leq\sum_{i,j,k=1}^{n}\frac{1}{\lambda_{i}\lambda_{j}\lambda_{k}}|\varphi_{i\overline{j}k}|^{2}=\mathcal{S}\leq\frac{Ce^{Ae^{B\sigma^{-\lambda}}}}{t^{5/2}},$$

and using (3.33) we get

$$|\nabla\tilde{\omega}_{y}|^{2}_{\omega_{y}}=\sum_{i,j,k=1}^{n-m}|\varphi_{i\overline{j}k}|^{2}\leq t^{1/2}Ce^{(A+3C_{0})e^{B\sigma^{-\lambda}}},$$

and this is (2.11).