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

Let $S$ be a closed surface of genus at least $2$. For each maximal representation $rho: pi_1(S)rightarrowmathsf{Sp}(4,mathbb{R})$ 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 Labouries recent results on Hitchin components, these bundles are not vector bundles.