Area Rigidity for the Regular Representation of Surface Groups

TL;DR

This paper characterizes area-minimizing maps from the universal cover of a surface to a Hilbert sphere under a regular representation, classifies minimal surfaces with negative curvature, and extends classical results.

Contribution

It provides a complete characterization of equivariantly area-minimizing maps for regular representations and classifies minimal surfaces in Hilbert spheres with negative curvature.

Findings

All equivariantly area-minimizing maps are characterized.

Classified all minimal surfaces in Hilbert spheres with constant negative Gaussian curvature.

Extended classical results of Calabi, Kenmotsu, and Bryant.

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.