Maximal $${\mathsf {Sp}}(4,{\mathbb {R}})$$ surface group representations, minimal immersions and cyclic surfaces

Abstract

Let S be a closed surface of genus at least 2. For each maximal representation $$\rho : \pi _1(S){\rightarrow }{\mathsf {Sp}}(4,{\mathbb {R}})$$ 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 Teichmuller space. Unlike Labourie’s recent results on Hitchin components, these bundles are not vector bundles.