Maximal Sp(4,R) surface group representations, minimal immersions and cyclic surfaces

Abstract

Let $S$ be a closed surface of genus at least $2$. For each maximal representation $\rho :{\pi }_1(S)\to \mathsf{Sp}(4,\mathbb{R})$ in one of the $2g-3$ exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric space $\mathsf{Sp}(4,\mathbb{R})/\mathsf{U}(2)$ is a minimal immersion. Using a Higgs bundle parameterization of these components, we give a mapping class group invariant parameterization of such components as fiber bundles over Teichmüller space. Unlike Labourie's recent results on Hitchin components, these bundles are not vector bundles.

37 pages, comments welcome, v2 mistakes in the proof Theorem 3.8 corrected

References in corpus (1)

• G2 geometry and integrable systems