Abstract : Higgs bundle theory has proven to be a power tool in
describing character varieties of surface group representations.
However, it is difficult in general to understand geometric properties
of a given representation by looking at its corresponding Higgs
bundle, because the correspondence involves solving a system of PDEs.
In recent work with Brian Collier and Jeremy Toulisse, we derive
from Higgs bundle theory refined geometric properties of representations
into a Hermitian Lie group of rank 2 with maximal Toledo invariant.
The key fact is that Higgs bundles provide us with the existence of
a minimal surface in a pseudo-Riemannian G-homogeneous space, whose
geometry is easier to study than that of a Riemannian symmetric
space of a Lie group