@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} }