Integral gradient estimates on a closed surface

Abstract

Let $(\hat{I}\pounds ,g)$ be a closed Riemann surface, and let $u$ be a weak solution to equation \[ - Δ_g u = μ, \] where $\hat{I}¼$ is a signed Radon measure. We aim to establish $L^p$ estimates for the gradient of $u$ that are independent of the choice of the metric $g$. This is particularly relevant when the complex structure approaches the boundary of the moduli space. To this end, we consider the metric $g^{\prime }=e^{2u}g$ as a metric of bounded integral curvature. This metric satisfies a so-called quadratic area bound condition, which allows us to derive gradient estimates for $g^{\prime }$ in local conformal coordinates. From these estimates, we obtain the desired estimates for the gradient of $u$.