Concave Foliated Flag Structures and the ${\text{SL}}_3(\mathbb{R})$ Hitchin Component

Abstract

We give a geometric characterization of flag geometries associated to Hitchin representations in ${\text{SL}}_3(\mathbb{R})$. Our characterization is based on distinguished invariant foliations, similar to those studied by Guichard-Wienhard in ${\text{PSL}}_4(\mathbb{R})$. We connect to the dynamics of Hitchin representations by constructing refraction flows for all positive roots in general ${\mathfrak{sl}}_n(\mathbb{R})$ in our setting. For $n=3$, leaves of our one-dimensional foliations are flow-lines. One consequence is that the highest root flows are $C^{1+\alpha }$.

arXiv comment43 pages, 7 figures. Comments welcome!