On conformal planes of finite area (Q6060827)

A real-valued function \(u\) in the Euclidean plane is a supersolution to the Liouville equation \(\Delta v+e^{2v}=0\) if \(\Delta u+e^{2u}\leq0\). Let \(X_{u}=\left(\mathbb{R}^{2},e^{2u}\delta\right) \) be a conformal plane, where \(\delta\) is the Euclidean metric in \(\mathbb{R}^{2}\). In [Math. Z. 305, No. 3, Paper No. 40, 30 p. (2023; Zbl 1525.35131)], \textit{C. Gui} and \textit{Q. Li} obtained results relating diameter estimates for \(X_{u}\) with the finite conformal area, conformal area estimates, and others. In addition, Gui and Li posed several problems related to their results (Questions 8.1, 8.2, 8.3 and 8.7). In the paper under review, the author discusses solutions of these questions (Propositions 1.1, 1.2 and Corollary 2.1).