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

TL;DR

This paper provides a geometric characterization of flag geometries linked to Hitchin representations in SL_3(R), introducing invariant foliations and analyzing their dynamics, including flow properties and regularity.

Contribution

It introduces a new geometric characterization of flag structures for SL_3(R) Hitchin representations using invariant foliations and constructs refraction flows for all positive roots.

Findings

Flow-lines form the leaves of foliations in the SL_3(R) case.

Highest root flows are shown to be C^{1+α} smooth.

The work connects geometric structures with dynamical properties of Hitchin representations.

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 $C^{1+\alpha}$.