Cyclic surfaces and Hitchin components in rank 2

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.