Integral gradient estimates on a closed surface
Yuxiang Li, Rongze Sun
TL;DR
We address gradient estimates for weak solutions of $-\Delta_g u = \mu$ on closed surfaces in a way that is independent of the metric by passing to the conformal metric $g'=e^{2u}g$ with bounded integral curvature. The core method combines Brezis–Merle theory, the BIC/ Alexandrov framework, and Li–Tang–Sun type intrinsic gradient bounds to obtain uniform $L^p$ control of $\nabla u$ on disks and, via collar analysis, on collars and entire constant-curvature surfaces. The main results bound $E_p(u)$ and $E_{a,p}(u)$ in terms of $|\mu|(\Sigma)$, with corollaries giving precise gradient bounds on flat and hyperbolic surfaces, including torus and collar regions. This work enables gradient control in degenerating conformal classes, aiding compactness arguments in variational problems and the analysis of Green's functions and Poisson data on surfaces.
Abstract
Let $(Σ, g)$ be a closed Riemann surface, and let $u$ be a weak solution to equation \[ - Δ_g u = μ, \] where $μ$ 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 $u$.