For S a closed surface of genus at least 2, Hitchin representations from pi_1(S) to SL(n,R) naturally generalize Fuchsian representations to SL(2,R), which are equivalent to hyperbolic structures on S. Labourie proved that every Hitchin representation comes with an invariant minimal surface in the corresponding symmetric space, and he conjectured that uniqueness holds as well.

In this talk, we'll explain how we used Higgs bundles to produce large area minimal surfaces that give counterexamples to Labourie's conjecture (for all n at least 4), and we'll overview recent and related advances in the theory of Higgs bundles at high energy. This represents joint work with Peter Smillie, plus a new work in progress.