Unstable minimal surfaces in symmetric spaces of non-compact type

Abstract

We prove that if $\Sigma$ is a closed surface of genus at least 3 and $G$ is a split real semisimple Lie group of rank at least $3$ acting faithfully by isometries on a Riemannian symmetric space $N$, then there exists a Hitchin representation $\rho \pi_1(\Sigma)\to G$ and a $\rho$-equivariant unstable minimal map from the universal cover of $\Sigma$ to $N$. This follows from a new lower bound on the index of high energy minimal maps into an arbitrary symmetric space of non-compact type. Taking $G=\mathrm{PSL}(n,\mathbb{R})$,
$n\geq 4$, this disproves the Labourie Conjecture.