标题： 积分梯度估计在闭曲面上 显示英文标题

标题： Integral gradient estimates on a closed surface

摘要： 设$(\Sigma, g)$为一个闭合的黎曼曲面，令$u$为方程\[ - \Delta_g u = \mu, \]的弱解，其中$\mu$是一个带符号的Radon测度。 我们旨在建立$L^p$对于$u$梯度的估计，这些估计与度量$g$的选择无关。 当复结构接近模空间边界时，这尤其相关。 为此，我们考虑度量$g' = e^{2u} g$作为有界积分曲率的度量。 这个度量满足所谓的二次面积界条件，这使我们能够在局部共形坐标中推导出$g'$的梯度估计。 从这些估计中，我们得到了$u$梯度的期望估计。 显示英文摘要

摘要： 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 $u$.