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 $\rho: \pi_1(S)\rightarrow\mathsf{Sp}(4,\mathbb{R})$ 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 Teichm\"uller space. Unlike Labourie's recent results on Hitchin components, these bundles are not vector bundles.