Trudinger–Moser Inequalities on a Closed Riemann Surface with a Symmetric Conical Metric

This is a continuation of our previous work (Ann. Sc. Norm. Super. Pisa Cl. Sci., 20, 1295–1324, 2020). Let (Σ, g) be a closed Riemann surface, where the metric g has conical singularities at finite points. Suppose G is a group whose elements are isometries acting on (Σ, g). Trudinger–Moser inequalities involving G are established via the method of blow-up analysis, and the corresponding extremals are also obtained. This extends previous results of Chen (Proc. Amer. Math. Soc., 1990), Iula–Manicini (Nonlinear Anal., 2017), and the authors (2020).