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Volume 37, Issue 1
Completion of $\mathbb{R}^2$ with a Conformal Metric as a Closed Surface

Changfeng Gui & Qinfeng Li

Anal. Theory Appl., 37 (2021), pp. 59-73.

Published online: 2021-04

[An open-access article; the PDF is free to any online user.]

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  • 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.

  • AMS Subject Headings

35B08, 35J15, 35J61, 53C18

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COPYRIGHT: © Global Science Press

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@Article{ATA-37-59, author = {Gui , Changfeng and Li , Qinfeng}, title = {Completion of $\mathbb{R}^2$ with a Conformal Metric as a Closed Surface}, journal = {Analysis in Theory and Applications}, year = {2021}, volume = {37}, number = {1}, pages = {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.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2021.pr80.10}, url = {http://global-sci.org/intro/article_detail/ata/18764.html} }
TY - JOUR T1 - Completion of $\mathbb{R}^2$ with a Conformal Metric as a Closed Surface AU - Gui , Changfeng AU - Li , Qinfeng JO - Analysis in Theory and Applications VL - 1 SP - 59 EP - 73 PY - 2021 DA - 2021/04 SN - 37 DO - http://doi.org/10.4208/ata.2021.pr80.10 UR - https://global-sci.org/intro/article_detail/ata/18764.html KW - Gaussian curvature, conformal geometry, semilinear equations, entire solutions. AB -

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.

Gui , Changfeng and Li , Qinfeng. (2021). Completion of $\mathbb{R}^2$ with a Conformal Metric as a Closed Surface. Analysis in Theory and Applications. 37 (1). 59-73. doi:10.4208/ata.2021.pr80.10
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