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Integral gradient estimates on a closed surface

Part of: Global differential geometry

Published online by Cambridge University Press:  10 October 2025

Yuxiang Li  and
Rongze Sun
Yuxiang Li
Affiliation:
Tsinghua University , China e-mail: liyuxiang@tsinghua.edu.cn
Rongze Sun*
Affiliation:
Tsinghua University , China e-mail: liyuxiang@tsinghua.edu.cn
*
e-mail: srz20@mails.tsinghua.edu.cn
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Abstract

Let $(\Sigma , g)$ be a closed Riemann surface, and let u be a weak solution to the equation

$$\begin{align*}- \Delta_g u = \mu, \end{align*}$$
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.

Keywords

Integral gradient estimateflat torushyperbolic surfaces

MSC classification

Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 16
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The first author is partially supported by the National Key R&D Program of China 2022YFA1005400.

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