Cyclic surfaces and Hitchin components in rank 2

TL;DR

This paper establishes a unique correspondence between Hitchin representations in rank 2 groups and equivariant minimal surfaces, providing a new parametrization of Hitchin components via Hermitian bundles over Teichmüller space.

Contribution

It introduces a novel geometric approach linking Hitchin representations to minimal surfaces and Hermitian bundles, extending some constructions to higher rank cyclic bundles.

Findings

Existence of unique equivariant minimal surfaces for Hitchin representations in rank 2.

Parametrization of Hitchin components by Hermitian bundles over Teichmüller space.

Partial extensions of the construction to higher rank cyclic bundles.

Abstract

We prove that given a Hitchin representation in a real split rank 2 group $\mathsf G_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin components by a Hermitian bundle over Teichm\"uller space. The proof goes through introducing holomorphic curves in a suitable bundle over the symmetric space of $\mathsf G_0$. Some partial extensions of the construction hold for cyclic bundles in higher rank.