Let $G_{\epsilon,t}(x,y)$ denote Green’s function for the Laplacian $\Delta_{\epsilon,t}$ associated to the metric $\omega_{\epsilon,t}$, normalised so that $\int_{X}G_{\epsilon,t}(x,y)\,\omega^{2}_{\epsilon,t}(x)=0$. Then, for $\epsilon$ sufficiently small and any $t\in[0,1]$,

$$G_{\epsilon,t}(x,y)\geq-A\epsilon^{-4},$$

for some constant $A$ independent of $\epsilon$ and $t$.