Geometric Structures for $G_2^{\mathrm{\prime }}$-Surface Group Representations

Let $S$ be a closed surface of genus $g\ge 2$. We construct locally homogeneous geometric structures on closed 5-manifolds fibering over $S$, modeled on the two partial flag manifolds ${\mathrm{E}\mathrm{i}\mathrm{n}}^{2,3}$ and ${\mathrm{P}\mathrm{h}\mathrm{o}}^{\times }$ of the split real form ${\mathrm{G}}_2^{\mathrm{\prime }}$ of the complex exceptional Lie group ${\mathrm{G}}_2^{\mathbb{C}}$. To this end, we consider two families of representations ${\pi }_{1}S\to {\mathrm{G}}_2^{\mathrm{\prime }}$ constructed via the non-abelian Hodge correspondence from cyclic Higgs bundles, one associated with each ${\mathrm{G}}_2^{\mathrm{\prime }}$-partial flag manifold. Each family includes ${\mathrm{G}}_2^{\mathrm{\prime }}$-Hitchin representations, but is much more general. For each representation of the first family, the $\beta$-bundles, we construct $({\mathrm{G}}_2^{\mathrm{\prime }},{\mathrm{E}\mathrm{i}\mathrm{n}}^{2,3})$-geometric structures on ${\mathrm{E}\mathrm{i}\mathrm{n}}^{2,1}$-fiber bundles over $S$, and for Hodge bundles in the second family we construct $({\mathrm{G}}_2,{\mathrm{P}\mathrm{h}\mathrm{o}}^{\times })$-geometric structures on $(\mathbb{R}{\mathbb{P}}^2\times {\mathbb{S}}^1)$-bundles over $S$. In the case of ${\mathrm{G}}_2^{\mathrm{\prime }}$-Hitchin Hodge bundles, which belong to both families, we show the image of the developing map of the respective geometric structures is exactly the domain of discontinuity defined by Guichard-Wienhard and Kapovich-Leeb-Porti. Each construction can be interpreted as converting a family of equivariant $J$-holomorphic curves in the pseudosphere ${\mathbb{S}}^{2,4}$ into geometric structures on fiber bundles $M\to S$. The approach used to build geometric structures, namely \emph{moving bases of pencils}, gives a unified description of analytic geometric structures constructions using Higgs bundles and harmonic maps in rank two.