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 - <p style="text-align: justify;">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.</p>