(untitled) (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 R^2, if the total Gaussian curvature is 4π, the conformal area of R^2 is finite and the Gaussian curvature is bounded, then R^2 is a compact C^1,α surface after completion at ∞, for any α (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.