Completion of $\mathbb{R}^2$ with a Conformal Metric as a Closed Surface

Year:    2021

Author:    Changfeng Gui, Qinfeng Li

Analysis in Theory and Applications, Vol. 37 (2021), Iss. 1 : pp. 59–73

Abstract

In this paper, we obtain some asymptotic behavior results for solutions to the prescribed Gaussian curvature equation. Moreover, we prove that under a conformal metric in $\mathbb{R}^2$, if the total Gaussian curvature is $4\pi$, the conformal area of $\mathbb{R}^2$ is finite and the Gaussian curvature is bounded, then $\mathbb{R}^2$ is a compact $C^{1,\alpha}$ surface after completion at $\infty$, for any $\alpha \in (0,1)$. If the Gaussian curvature has a Hölder decay at infinity, then the completed surface is $C^2$. For radial solutions, the same regularity holds if the Gaussian curvature has a limit at infinity.

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/ata.2021.pr80.10

Analysis in Theory and Applications, Vol. 37 (2021), Iss. 1 : pp. 59–73

Published online:    2021-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    15

Keywords:    Gaussian curvature conformal geometry semilinear equations entire solutions.

Author Details

Changfeng Gui

Qinfeng Li

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    Lytchak, Alexander

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    https://doi.org/10.1007/s00209-023-03381-9 [Citations: 1]