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Integral gradient estimates on a closed surface
- Part of
-
- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 10 October 2025, pp. 1-16
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- Article
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Let
$(\Sigma , g)$ be a closed Riemann surface, and let u be a weak solution to the equation where
$$\begin{align*}- \Delta_g u = \mu, \end{align*}$$
$\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.
