  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.
