On Foliations in $\text{PSL}(4,\mathbb{R})$-Teichmüller Theory

We carry out a detailed study of the structure of domains of discontinuity ${\mathrm{\Omega }}_{\rho }$ in ${\mathbb{R}\mathbb{P}}^3$ of ${\text{PSL}}_4(\mathbb{R})$-Hitchin representations $\rho$. We then prove the foliated component ${\mathrm{\Omega }}_{\rho }^1$ of ${\mathrm{\Omega }}_{\rho }$ has exactly two group-invariant foliations by properly embedded projective line segments and has a unique foliation by properly embedded convex domains in projective planes. This gives a finiteness counterpart to work of Guichard and Wienhard. We also prove analogues for the non-foliated component ${\mathrm{\Omega }}_{\rho }^2$ and deduce a rigidity of projective equivalences of properly convex foliated projective structures on unit tangent bundles of surfaces.