Trudinger-Moser inequalities on a closed Riemannian surface with the action of a finite isometric group

Let $(\mathrm{\Sigma },g)$ be a closed Riemannian surface, $W^{1,2}(\mathrm{\Sigma },g)$ be the usual Sobolev space, $\mathbf{\text{G}}$ be a finite isometric group acting on $(\mathrm{\Sigma },g)$, and ${\mathcal{H}}_{\mathbf{\text{G}}}$ be a function space including all functions $u\in W^{1,2}(\mathrm{\Sigma },g)$ with ${\int }_{\mathrm{\Sigma }}ud{v}_g=0$ and $u(\sigma (x))=u(x)$ for all $\sigma \in \mathbf{\text{G}}$ and all $x\in \mathrm{\Sigma }$. Denote the number of distinct points of the set $\{\sigma (x):\sigma \in \mathbf{\text{G}}\}$ by $I(x)$ and $\mathrm{\ell }={\inf }_{x\in \mathrm{\Sigma }}I(x)$. Let ${\lambda }_1^{\mathbf{\text{G}}}$ be the first eigenvalue of the Laplace-Beltrami operator on the space ${\mathcal{H}}_{\mathbf{\text{G}}}$. Using blow-up analysis, we prove that if $\alpha <{\lambda }_1^{\mathbf{\text{G}}}$ and $\beta \le 4\pi \mathrm{\ell }$, then there holds $$\underset{u\in {\mathcal{H}}_{\mathbf{\text{G}}},{\int }_{\mathrm{\Sigma }}\mathrm{\mid }{\mathrm{\nabla }}_{g}u{\mathrm{\mid }}^{2}d{v}_g-\alpha {\int }_{\mathrm{\Sigma }}{u}^{2}d{v}_g\le 1}{\sup }{\int }_{\mathrm{\Sigma }}{e}^{\beta u^2}d{v}_g<\mathrm{\infty };$$ if $\alpha <{\lambda }_1^{\mathbf{\text{G}}}$ and $\beta >4\pi \mathrm{\ell }$, or $\alpha\geq \lambda_1^$\mathbf{\text{G}}$$and$\beta>0$,thentheabovesupremumisinfinity;if$\alpha<\lambda_1^$\mathbf{\text{G}}$$and$\beta\leq 4\pi\ell$, then the above supremum can be attained. Moreover, similar inequalities involving higher order eigenvalues are obtained. Our results partially improve original inequalities of J. Moser \cite{Moser}, L. Fontana \cite{Fontana} and W. Chen \cite{Chen-90}.