Geometric Structures for $G_2'$-Surface Group Representations

Abstract

Let $S$ be a closed surface of genus $g \geq 2$. We construct locally homogeneous geometric structures on closed 5-manifolds fibering over $S$, modeled on the two partial flag manifolds $\mathrm{Ein}^{2,3}$ and $\mathrm{Pho}^\times$ of the split real form $\mathrm{G}_2'$ of the complex exceptional Lie group $\mathrm{G}_2^{\mathbb{C}}$. To this end, we consider two families of representations $\pi_1S\rightarrow \mathrm{G}_2'$ constructed via the non-abelian Hodge correspondence from cyclic Higgs bundles, one associated with each $\mathrm{G}_2'$-partial flag manifold. Each family includes $\mathrm{G}_2'$-Hitchin representations, but is much more general. For each representation of the first family, the $\beta$-bundles, we construct $(\mathrm{G}_2', \mathrm{Ein}^{2,3})$-geometric structures on $\mathrm{Ein}^{2,1}$-fiber bundles over $S$, and for Hodge bundles in the second family we construct $(\mathrm{G}_2, \mathrm{Pho}^\times)$-geometric structures on $(\mathbb{R} \mathbb{P}^2\times \mathbb{S}^1)$-bundles over $S$. In the case of $\mathrm{G}_2'$-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 $M \rightarrow S$. 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.