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

TL;DR

This paper establishes refined Trudinger-Moser inequalities on closed Riemannian surfaces with symmetry group actions, identifying sharp conditions for finiteness and attainability of the associated exponential integrals, and extends previous results.

Contribution

It introduces new inequalities involving the action of finite isometric groups and higher eigenvalues, improving upon classical results by Moser, Fontana, and Chen.

Findings

Identifies sharp bounds for exponential integrals under symmetry constraints.

Proves attainability of the supremum in certain parameter regimes.

Extends inequalities to involve higher order eigenvalues.

Abstract

Let $(\Sigma,g)$ be a closed Riemannian surface, $W^{1,2}(\Sigma,g)$ be the usual Sobolev space, $\textbf{G}$ be a finite isometric group acting on $(\Sigma,g)$, and $\mathscr{H}_\textbf{G}$ be a function space including all functions $u\in W^{1,2}(\Sigma,g)$ with $\int_\Sigma udv_g=0$ and $u(\sigma(x))=u(x)$ for all $\sigma\in \textbf{G}$ and all $x\in\Sigma$. Denote the number of distinct points of the set $\{\sigma(x): \sigma\in \textbf{G}\}$ by $I(x)$ and $\ell=\inf_{x\in \Sigma}I(x)$. Let $\lambda_1^\textbf{G}$ be the first eigenvalue of the Laplace-Beltrami operator on the space $\mathscr{H}_\textbf{G}$. Using blow-up analysis, we prove that if $\alpha<\lambda_1^\textbf{G}$ and $\beta\leq 4\pi\ell$, then there holds $$\sup_{u\in\mathscr{H}_\textbf{G},\,\int_\Sigma|\nabla_gu|^2dv_g-\alpha \int_\Sigma u^2dv_g\leq 1}\int_\Sigma e^{\beta u^2}dv_g<\infty;$$ if $\alpha<\lambda_1^\textbf{G}$ and $\beta>4\pi\ell$, or $\alpha\geq \lambda_1^\textbf{G}$ and $\beta>0$, then the above supremum is infinity; if $\alpha<\lambda_1^\textbf{G}$ 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}.