$G_2^{\prime }$-Slodowy Slices I: Geometric Structures

Abstract

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 ${\mathrm{Ein}}^{2,3}$ and ${\mathrm{Pho}}^{\times }$ of the split real form ${\mathrm{G}}_2^{\prime }$ of the complex exceptional Lie group ${\mathrm{G}}_2^{\mathbb{C}}$. To this end, we consider two families of representations ${\pi }_{1}S\to {\mathrm{G}}_2^{\prime }$ constructed via the non-abelian Hodge correspondence from cyclic Higgs bundles, one associated with each ${\mathrm{G}}_2^{\prime }$-partial flag manifold. Each family includes ${\mathrm{G}}_2^{\prime }$-Hitchin representations, but is much more general. From the Higgs bundles of the first family, called $\beta$-bundles, we construct $({\mathrm{G}}_2^{\prime },{\mathrm{Ein}}^{2,3})$-geometric structures on ${\mathrm{Ein}}^{2,1}$-fiber bundles over $S$, and from Hodge bundles in the second family, called $\alpha$-bundles, we construct $({\mathrm{G}}_2^{\prime },{\mathrm{Pho}}^{\times })$-geometric structures on $({\mathbb{RP}}^2\times {\mathbb{S}}^1)$-bundles over $S$. In the case of ${\mathrm{G}}_2^{\prime }$-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 ${\hat{\mathbb{S}}}^{2,4}$ 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 prior analytic geometric structures constructions using Higgs bundles and harmonic maps.

Revised title and expositional improvements. 78 pages + 10 pages appendices. 4 figures, 2 tables