An equivariant minimal surface in $n$-dimensional complex hyperbolic space $\CH{n}$ is a triple consisting of a minimal immersion (possibly branched) of the Poincar\'e disc into $\CH{n}$ and two representations of the fundamental
group of a closed oriented surface: one a Fuchsian representation into the isometry group of the disc, and the other a reductive representation into the isometry group of $\CH{n}$. We require the minimal immersion to be equivariant
with respect to these two representations.
We see how such surfaces have both a corresponding Higgs bundle and a harmonic sequence via the non-abelian Hodge correspondence. This provides an original application of global harmonic sequence theory. In order to do this, we adapt known results about the moduli space of Higgs bundles, develop both the local and global theory for $\CH{n}$ harmonic sequences and apply the latter to the former in a new context.
We consider the critical submanifolds fixed under the $\bb{C}^*$-action of the Higgs field. By considering their invariants, we fully classify these subspaces of Hodge bundles, and make clear links back to other geometric measures such as curvature. We also establish necessary conditions for stability of these as Higgs bundles.
Finally, we see how the rest of the moduli space can be understood in terms of a Hodge bundle and an extension, before linking this back to invariants from harmonic sequences. We consider the limits of the $\bb{C}^*$-action, describing some explicitly. Applying the concept of the isotropy order from harmonic sequences to Higgs bundles, we end by examining the existence of a Higgs bundle of a given isotropy order. The main results prove the existence of $\CH{3}$ Higgs bundles of all possible isotropy order and then allow us to find topological conditions for the existence of $\CH{n}$ Higgs bundles of a given isotropy order under certain assumptions.
