Unstable minimal surfaces in symmetric spaces of non-compact type

TL;DR

The paper constructs unstable minimal surfaces associated with Hitchin representations in symmetric spaces of non-compact type, providing new lower bounds on the index of minimal maps and disproving a conjecture in the field.

Contribution

It introduces a novel lower bound on the index of high energy minimal maps and demonstrates the existence of unstable minimal surfaces for certain Hitchin representations, disproving Labourie's conjecture.

Findings

Existence of unstable minimal surfaces for specific Hitchin representations

New lower bound on the index of minimal maps into symmetric spaces

Disproof of Labourie's conjecture for certain groups

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 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.