The moduli spaces of equivariant minimal surfaces in $\mathbb{RH}^3$ and $\mathbb{RH}^4$ via Higgs bundles

TL;DR

This paper introduces a new framework for understanding the moduli spaces of equivariant minimal surfaces in hyperbolic spaces using Higgs bundles, providing detailed descriptions for dimensions 3 and 4.

Contribution

It defines the moduli space of equivariant minimal immersions into non-compact symmetric spaces and links it to Higgs bundle parametrizations, especially for $ ext{RH}^3$ and $ ext{RH}^4$.

Findings

Higgs bundles effectively parametrize the moduli spaces.

Classical invariants are incorporated into the Higgs bundle framework.

Detailed structure of moduli spaces for $ ext{RH}^3$ and $ ext{RH}^4$ is elucidated.

Abstract

In this article we introduce a definition for the moduli space of equivariant minimal immersions of the Poincar\'e disc into a non-compact symmetric space, where the equivariance is with respect to representations of the fundamental group of a compact Riemann surface of genus at least two. We then study this moduli space for the non-compact symmetric space $\mathbb{RH}^n$ and show how $SO_0(n,1)$-Higgs bundles can be used to parametrise this space, making clear how the classical invariants (induced metric and second fundamental form) figure in this picture. We use this parametrisation to provide details of the moduli spaces for $\mathbb{RH}^3$ and $\mathbb{RH}^4$, and relate their structure to the structure of the corresponding Higgs bundle moduli spaces.