Integral gradient estimates on a closed surface
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909845676883968 |
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| author | Li, Yuxiang Sun, Rongze |
| author_facet | Li, Yuxiang Sun, Rongze |
| contents | 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 $u$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12790 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integral gradient estimates on a closed surface Li, Yuxiang Sun, Rongze Differential Geometry Analysis of PDEs 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 $u$. |
| title | Integral gradient estimates on a closed surface |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2507.12790 |