There are constants $A,B,C$ that depend only on the fixed data, so that on $X\backslash S$ and for any $0<t\leq 1$ we have

(2.9)

$$\frac{t}{Ce^{Ae^{B\sigma^{-\lambda}}}}\omega_{X}\leq\tilde{\omega}_{t}\leq Ce^{Ae^{B\sigma^{-\lambda}}}\omega_{X},$$

where $\sigma$ is defined by (2.2). In particular the Laplacian $\Delta_{\omega_{X}}\varphi_{t}$ is bounded uniformly on compact sets of $X\backslash S$, independent of $t$.