G2 geometry and integrable systems

Abstract

We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real simple Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspond to a form of the affine Toda equations. In each case we relate cyclic Higgs bundles to geometric structures on the surface. We elucidate the geometry of generic 2-plane distributions in 5 dimensions, relating it to a parabolic geometry associated to the split real form of $G_2$ and a conformal geometry with holonomy in $G_2$. We prove the distribution is the bundle of maximal isotropics corresponding to the annihilator of a spinor satisfying the twistor-spinor equation. We study the moduli space of coassociative submanifolds of a $G_2$-manifold with an aim towards understanding coassociative fibrations. We consider coassociative fibrations where the fibres are orbits of a $T^4$-action of isomorphisms and prove a local equivalence to minimal 3-manifolds in $R^{3,3}\cong H^2(T^4,\mathbb{R})$ with positive induced metric.

Thesis 149 pages, minor corrections

References in corpus (8)

• Differential equations and conformal structures
• Variational Approach to Differential Invariants of Rank 2 Vector Distributions
• Duality and Fibrations on G_2 Manifolds
• Ricci-corrected derivatives and invariant differential operators
• Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds - Characterization and Killing-Field Decomposition
• On Nurowski's conformal structure associated to a generic rank two distribution in dimension five
• Coassociative K3 fibrations of compact G_2-manifolds
• Inclusions between parabolic geometries

Cited by in corpus (18)

• Asymptotics of certain families of Higgs bundles in the Hitchin component
• An Introduction to Higgs Bundles via Harmonic Maps
• Higgs bundles and geometric structures on manifolds
• Maximal Sp(4,R) surface group representations, minimal immersions and cyclic surfaces
• Planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4,\mathbb{R})$-symmetric space
• Convex RP^2 Structures and Cubic Differentials under Neck Separation
• Minimal Lagrangian connections on compact surfaces
• Geodesic Coordinates for the Pressure Metric at the Fuchsian Locus
• Isolated singularities of Toda equations and cyclic Higgs bundles
• ADS 3-manifolds and Higgs bundles
• Minimal surfaces for Hitchin representations
• Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces
• Projective Structures with (Quasi-)Hitchin Holonomy
• On diagonal pluriharmonic metrics of $G$-Higgs bundles
• Length spectrum compactification of the ${\mathrm{SO}}_0(2,3)$-Hitchin component
• Almost strict domination and anti-de Sitter 3-manifolds
• The conformal limit and projective structures
• Harmonic maps for Hitchin representations