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 inwhich 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.

Asymptotics of certain families of Higgs bundles in the Hitchin component

Using Hitchins parameterization of the Hitchin-Teichmuller component of the $SL(n,mathbb{R})$ representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the $SL(n,mathbb{R})$-Hitchincomponent, we study the asymptotics of the Hermitian metric solving the Higgs bundle equations. This analysis is used to estimate the asymptotics of the corresponding family of flat connections as we scale the differentials by a real parameter. We consider Higgs fields that have only one holomorphic differential $q_n$ of degree $n$ or $q_{n-1}$ of degree $n-1.$ We also study the asymptotics of the associated family of equivariant harmonic maps to the symmetric space $SL(n,mathbb{R})/SO(n,mathbb{R})$ and relate it to recent work of Katzarkov, Noll, Pandit and Simpson.