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 $\ell=\inf_{x\in \Sigma}I(x)$... (read more)

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