Non-uniqueness of minimal surfaces in locally symmetric spaces

If G is a split real Lie group of rank 2, for instance SL(3,R), and S is a closed surface of genus at least 2, then Labourie showed that every Hitchin representation of pi_1(S) into G admits a unique equivariant minimal surface. As Labourie pointed out, this lets you parametrise the space of Hitchin representations by the total space of a vector bundle over the Teichmuller space of S. He conjectured that uniqueness should hold more generally, at least for all SL(n,R).

In joint work with Nathaniel Sagman, we show that for any split G of rank at least 3, and for any S, there is a Hitchin representation with two distinct equivariant minimal surfaces, disproving Labourie’s conjecture. I will explain our construction, which starts from minimal surfaces in R^3, and what new questions this raises.