Completion of $\mathbb{R}^2$ with a Conformal Metric as a Closed Surface (Analysis in Theory and Applications, 2022)
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 $4π$, the conformal area of $\mathbb{R}^2$ is finite and the Gaussian curvature is bounded, then $\mathbb{R}^2$ is a compact $C^{1,α}$ surface after completion at $∞$, for any $α \in (0,1)$. 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.
Keywords: Gaussian curvature; conformal geometry; semilinear equations; entire solutions.