https://arxiv.org/abs/2208.04885
Unstable minimal surfaces in symmetric spaces of non-compact type
Nathaniel Sagman, Peter Smillie
We prove that if Σ 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 symmetric space N, then there exists a Hitchin representation ρ:π1(Σ)→G and a ρ-equivariant unstable minimal map from the universal cover of Σ 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=PSL(n,ℝ), n≥4, this disproves the Labourie conjecture.