Question: Let $\Delta$ be the unit disc in $\mathbb{C}$ and $\rho(z)|dz|^2$ be a complete conformal metric on $\Delta$ where $\rho(z)$ is continuous on $\Delta$. Let $a$ be the infimum of $p (p>0)$ such that $\iint_\Delta |\rho(z)|^pdxdy=+\infty.$

I guess that $a\leq 1$. Of course, generally $a$ depends on the complete metric $\rho(z)$. For example, w.r.t. the Poincare metric, $a=\frac{1}{2}.$ Also, one may consider the infimum of $a$.

Note. We only assume that $\rho(z)$ is continuous and complete on the unit disc.