As $t$ goes to zero we have

$$\tilde{\omega}_{t}\to f^{*}\omega$$

in $C^{\infty}_{loc}(M\backslash S,\omega_{M})$, where $\omega=\omega_{N}+\sqrt{-1}\partial\overline{\partial}\varphi$ is a Kähler metric on $N\backslash f(S)$ with $\Ric(\omega)=\omega_{\rm WP}$ as in Theorem 1.1.