  Let $g=e^{2u}(dx^2+dy^2)$ be a conformal metric defined on the unit disk of
$\mathbf{C}$. We give an estimate of $\|\nabla
u\|_{L^{2,\infty}(D_\frac{1}{2})}$ when $\|K(g)\|_{L^1}$ is small and
$\frac{\mu(B_r^g(z),g)}{\pi r^2}<\Lambda$ for any $r$ and $z\in D_\frac{3}{4}$.
Then we will use this estimate to study the Gromov-Hausdorff convergence of a
conformal metric sequence with bounded $\|K\|_{L^1}$ and give some
applications.
