Cyclic surfaces and Hitchin components in rank 2

Abstract

We prove that given a Hitchin representation in a split real rank 2 group $\mathsf{G}_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrisation of the Hitchin component by a Hermitian bundle over Teichmüller 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.