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arXiv · 2508.19480

Area Rigidity for the Regular Representation of Surface Groups

Riccardo Caniato·Xingzhe Li·Antoine Song
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Abstract

Let $\tilde{\Sigma}$ be the universal cover of a closed surface $\Sigma$ of genus at least $2$. We characterize all equivariantly area-minimizing maps from $\tilde{\Sigma}$ to a Hilbert sphere, which are equivariant with respect to an isometric action of $\pi_1(\Sigma)$ weakly equivalent to the regular representation. As part of our proof, we classify all minimal surfaces in Hilbert spheres with constant negative Gaussian curvature. This builds on earlier results of E. Calabi, K. Kenmotsu, R. Bryant.

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Riccardo Caniato, Xingzhe Li, Antoine Song. 2025-08-26. Area Rigidity for the Regular Representation of Surface Groups. https://arxiv.org/abs/2508.19480

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