Integral gradient estimates on a closed surface
Yuxiang Li, Rongze Sun · Canadian Mathematical Bulletin · 2025
Abstract Let left parenthesis upper Sigma comma g right parenthesis $(\Sigma , g)$ ( Σ , g ) be a closed Riemann surface, and let u be a weak solution to the equation $$\begin{align*}- \Delta_g u = \mu, \end{align*}$$ − Δ g u = μ , where mu $\mu $ μ is a signed Radon measure. We aim to establish upper L Superscript p $L^p$ 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 equals e Superscript 2 u Baseline g $g' = e^{2u} g$ g ' = e 2 u 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 $g'$ g ' in local conformal coordinates. From these estimates, we obtain the desired estimates for the gradient of u .