Hitchin Functionals, h-Principles and Spectral Invariants
type: CreativeWork
description: This thesis investigates Hitchin functionals and $h$-principles for stable forms on oriented manifolds, with a special focus on $\mathrm{G}_2$ and $\widetilde{\mathrm{G}}_2$ 3- and 4-forms. Additionally, it introduces two new spectral invariants of torsion-free $\mathrm{G}_2$-structures. Part I begins by investigating an open problem posed by Bryant, $\textit{viz.}$ whether the Hitchin functional $\mathcal{H}_3$ on closed $\mathrm{G}_2$ 3-forms is unbounded above. Chapter 3 uses a scaling argument to obtain sufficient conditions for the functional $\mathcal{H}_3$ to be unbounded above and applies this result to prove the unboundedness above of $\mathcal{H}_3$ on two explicit examples of closed 7-manifolds with closed $\mathrm{G}_2$ 3-forms. Chapter 3 then proceeds to interpret this unboundedness geometrically, demonstrating an unexpected link between the functional $\mathcal{H}_3$ and fibrations, proving that the 'large volume limit' of $\mathcal{H}_3$ in each case corresponds to the adiabatic limit of a suitable fibration. The proof utilises a new, general collapsing result for singular fibrations between orbifolds, without assumptions on curvature, which is proved in Chapter 4. Chapter 5 broadens the focus of Part I to include the Hitchin functionals $\mathcal{H}_4$, $\widetilde{\mathcal{H}}_3$ and $\widetilde{\mathcal{H}}_4$ on closed $\mathrm{G}_2$ 4-forms, $\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms respectively. In its main result, Chapter 5 proves that $\mathcal{H}_4,\widetilde{\mathcal{H}}_3,\widetilde{\mathcal{H}}_4$ are always unbounded above and below (whenever defined), and also that $\mathcal{H}_3$ is always unbounded below (whenever defined). As scholia, the critical points of the functionals $\mathcal{H}_4$, $\widetilde{\mathcal{H}}_3$ and $\widetilde{\mathcal{H}}_4$ are shown to be saddle points, and initial conditions of the Laplacian coflow which cannot lead to convergent solutions are shown to be dense. Part I ends wi…
abstract: This thesis investigates Hitchin functionals and $h$-principles for stable forms on oriented manifolds, with a special focus on $\mathrm{G}_2$ and $\widetilde{\mathrm{G}}_2$ 3- and 4-forms. Additionally, it introduces two new spectral invariants of torsion-free $\mathrm{G}_2$-structures. Part I begins by investigating an open problem posed by Bryant, $\textit{viz.}$ whether the Hitchin functional $\mathcal{H}_3$ on closed $\mathrm{G}_2$ 3-forms is unbounded above. Chapter 3 uses a scaling argument to obtain sufficient conditions for the functional $\mathcal{H}_3$ to be unbounded above and applies this result to prove the unboundedness above of $\mathcal{H}_3$ on two explicit examples of closed 7-manifolds with closed $\mathrm{G}_2$ 3-forms. Chapter 3 then proceeds to interpret this unboundedness geometrically, demonstrating an unexpected link between the functional $\mathcal{H}_3$ and fibrations, proving that the 'large volume limit' of $\mathcal{H}_3$ in each case corresponds to the adiabatic limit of a suitable fibration. The proof utilises a new, general collapsing result for singular fibrations between orbifolds, without assumptions on curvature, which is proved in Chapter 4. Chapter 5 broadens the focus of Part I to include the Hitchin functionals $\mathcal{H}_4$, $\widetilde{\mathcal{H}}_3$ and $\widetilde{\mathcal{H}}_4$ on closed $\mathrm{G}_2$ 4-forms, $\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms respectively. In its main result, Chapter 5 proves that $\mathcal{H}_4,\widetilde{\mathcal{H}}_3,\widetilde{\mathcal{H}}_4$ are always unbounded above and below (whenever defined), and also that $\mathcal{H}_3$ is always unbounded below (whenever defined). As scholia, the critical points of the functionals $\mathcal{H}_4$, $\widetilde{\mathcal{H}}_3$ and $\widetilde{\mathcal{H}}_4$ are shown to be saddle points, and initial conditions of the Laplacian coflow which cannot lead to convergent solutions are shown to be dense. Part I ends wi…
author: Mayther, Laurence
language: eng
keywords: [split]G2-structures, Algebraic Topology, Associative Fibrations, Cobordisms, Contact Geometry, Convex Integration, Convex Integration with Avoidance, Differential Geometry, G2-Manifolds, G2-structures, Geometric Topology, Global Analysis, Gromov–Hausdorff Convergence, Hitchin Functionals, Homotopy Theory, h-Principles, Laplacian (Co)Flow, Metric Geometry, Orbifolds, SL(3;C)-structures, SL(3;R)^2-structures, Spectral Theory, Stable Forms, Symplectic Geometry, η-Invariants
license: https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/34200843-fc05-419f-b794-cf8729b4484c/download
license: https://www.rioxx.net/licenses/all-rights-reserved/
date published: 2023-06-30
date modified: 2026-07-22T22:24:23Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.104430