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

TL;DR

This paper extends Trudinger-Moser inequalities to closed Riemann surfaces with symmetric conical metrics, establishing new inequalities involving isometry groups and identifying extremal functions through blow-up analysis.

Contribution

It introduces new Trudinger-Moser inequalities on surfaces with conical singularities, incorporating symmetry groups, and determines extremal functions, extending prior results.

Findings

Established inequalities involving symmetry groups on conical surfaces

Identified extremal functions for the inequalities

Extended previous results to more general geometric settings

Abstract

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].