Concave foliated flag structures and the SL3(R) Hitchin component

We give a geometric characterization of flag geometries associated to Hitchin representations in ${\mathrm{SL}}_3\left(R\right)$. Our characterization is based on distinguished invariant foliations, similar to those studied by Guichard-Wienhard in ${\mathrm{PSL}}_4\left(R\right)$.

We connect to the dynamics of Hitchin representations by constructing refraction flows for all positive roots in general ${\mathrm{sl}}_n\left(R\right)$ in our setting. One consequence is that the highest root flows are $C^{1+\alpha }$. For $n=3$, leaves of our one-dimensional foliations are flow-lines.