Hostname: page-component-7f64f4797f-fz5kh Total loading time: 0 Render date: 2025-11-06T00:03:58.039Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

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

References

Attouch, H., Buttazzo, G., and Michaille, G., Variational analysis in Sobolev and BV spaces: Applications to PDEs and optimization. 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2014.Google Scholar
Bogachev, V., Measure theory. Vol. 1, Springer-Verlag, Berlin, 2007.Google Scholar
Brezis, H. and Merle, F., Uniform estimates and blow-up behavior for solutions of $-\varDelta u=V(x){e}^u$ in two dimensions . Commun. Partial Differ. Equ. 16(1991), 1223–1253.Google Scholar
Buser, P., Geometry and spectra of compact Riemann surfaces, Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2010.Google Scholar
Chen, J. and Li, Y., Uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature. Adv. Math. 479(2025), part B, Paper No. 110436, 73 pp.Google Scholar
Farkas, H. M. and Kra, I., Riemann surfaces. 2nd ed., Graduate Texts in Mathematics, 71, Springer-Verlag, New York, NY, 1992.Google Scholar
Kuwert, E. and Li, Y., ${W}^{2,2}$ -conformal immersions of a closed Riemann surface into ${\mathbb{R}}^n$ . Commun. Anal. Geom. 20(2012), no. 2, 313–340.Google Scholar
Laurain, P. and Rivière, T., Optimal estimate for the gradient of Green’s function on degenerating surfaces and applications . Commun. Anal. Geom. 26(2018), no. 4, 887–913.Google Scholar
Li, Y., Some remarks on Willmore surfaces embedded in ${\mathbb{R}}^3$ . J. Geom. Anal. 26(2016), no. 3, 2411–2424.Google Scholar
Li, Y., Sun, J., and Tang, H., Metrics on a surface with bounded total curvature . Int. Math. Res. Not. 17(2022), 13212–13245.Google Scholar
Li, Y., Wei, G., and Zhou, Z., John-Nirenberg radius and collapsing in conformal geometry . Asian J. Math. 24(2020), no. 5, 759–782.Google Scholar
Morrey, C. B., Mutiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, 130, Springer-Verlag New York, Inc., New York, NY, 1966.Google Scholar
Reshetnayk, Y. G., Isothermal coordinates on manifolds of bounded curvature I . Sibirsk. Mat. Zh. 1(1960), 88–116. (in Russian).Google Scholar
Reshetnayk, Y. G., Isothermal coordinates on manifolds of bounded curvature II . Sibirsk. Mat. Zh. 1(1960), 248–276. (in Russian).Google Scholar
Reshetnyak, Y. G., Two-dimensional manifolds of bounded curvature, geometry IV, 3-163, 245-250, Encyclopaedia of Mathematical Sciences, 70, Springer, Berlin, 1993.Google Scholar
Simon, L., Existence of surfaces minimizing the Willmore functional . Commun. Anal. Geom. 1(1993), no. 2, 281–326.Google Scholar
Troyanov, M., On Alexandrov surfaces with bounded integral curvature . In: F. Fillastre and D. Slutskiy (eds.), Reshetnyak’s theory of subharmonic metrics, Springer, Cham, 2023, pp. 9–34.Google Scholar