Integral gradient estimates on a closed surface

TL;DR

This paper develops metric-independent $L^p$ gradient estimates for solutions to Laplace equations on closed Riemann surfaces, especially near boundary points of the moduli space, using bounded integral curvature metrics.

Contribution

It introduces a novel approach to derive gradient estimates independent of the metric by employing metrics of bounded integral curvature and quadratic area bounds.

Findings

Established $L^p$ gradient estimates independent of the metric.

Applied quadratic area bounds to control gradient behavior.

Provided tools for analysis near moduli space boundaries.

Abstract

Let $(\Sigma, g)$ be a closed Riemann surface, and let $u$ be a weak solution to equation \[ - \Delta_g u = \mu, \] where $\mu$ 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 $g' = e^{2u} g$ 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 $g'$ in local conformal coordinates. From these estimates, we obtain the desired estimates for the gradient of $u$.