Trudinger-Moser inequalities on a closed Riemann surface with a symmetric conical metric

This is a continuation of our previous work [13]. Let $(\Sigma,g)$ be a closed Riemann surface, where the metric $g$ has conical singularities at finite points. Suppose $\mathbf{G}$ is a group whose elements are isometries acting on $(\Sigma,g)$. Trudinger-Moser inequalities involving $\mathbf{G}$ are established via the method of blow-up analysis, and the corresponding extremals are also obtained. This extends previous results of Chen [7], Iula-Manicini [21], and the authors [13].