Hitchin Functionals, h-Principles and Spectral Invariants

This thesis investigates Hitchin functionals and $h$-principles for stable forms on oriented manifolds, with a special focus on G₂ and \widetilde{G}₂ 3- and 4-forms. Additionally, it introduces two new spectral invariants of torsion-free G₂-structures. Part I begins by investigating an open problem posed by Bryant, viz. whether the Hitchin functional \mathcal{H}₃ on closed G₂ 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 G₂ 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 G₂ 4-forms, \widetilde{G}₂ 3-forms and \widetilde{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 with a short discussion of open questions, in Chapter 6. Part II investigates relative $h$-principles for closed, stable forms. After establishing some prerequisite algebraic results, Chapter 7 begins by proving that if a class of closed, stable forms satisfies the relative $h$-principle, then its corresponding Hitchin functional is automatically unbounded above. By utilising the technique of convex integration, Chapter 7 then obtains sufficient conditions for a class of closed, stable forms to satisfy the relative $h$-principle, a result which subsumes all previously established $h$-principles for closed stable forms. Until now, 12 of the 16 possible classes of closed stable forms have remained open questions with regard to the relative $h$-principle. In the main result of Part II, Chapters 7 and 8 prove the relative $h$-principle in 5 of these open cases. The remaining 7 cases are addressed in the final chapter of Part II, where it is conjectured that the relative $h$-principle holds in each case. Chapter 9 applies the $h$-principles established in this thesis to prove various results on the topological properties of closed \widetilde{G}₂, SL(3;\mathbb{C}) and SL(3;\mathbb{R})² forms. Firstly, it characterises which oriented 7-manifolds admit closed \widetilde{G}₂ forms, in the process introducing a new technique for proving the vanishing of natural cohomology classes on non-closed manifolds. Next, it introduces \widetilde{G}₂-cobordisms of closed SL(3;\mathbb{C}) and SL(3;\mathbb{R})² 3-forms and proves that homotopic forms are \widetilde{G}₂-cobordant. Additionally, Chapter 9 classifies SL(3;\mathbb{C}) 3-forms up to homotopy and provides a partial classification result on homotopy classes of SL(3;\mathbb{R})² 3-forms. Part II ends with a short discussion of open questions, in Chapter 10. Part III introduces and examines two new spectral invariants of torsion-free G₂-structures. Although the notion of an invariant is a central theme in geometry and topology, currently, there is only one known invariant of torsion-free G₂-structures: the $\overline{\nu}$-invariant of Crowley-Goette-Nordström. Part III defines two new invariants of torsion-free G₂-structures, termed μ₃- and μ₄-invariants, by regularising the classical notion of Morse index for the Hitchin functionals \mathcal{H}₃ and \mathcal{H}₄ at their critical points. In general, there is no known way to compute $\overline{\nu}$ for G₂-manifolds constructed via Joyce's `generalised Kummer construction'. Chapter 11 obtains closed formulae for μ₃ and μ₄ on the orbifolds used in Joyce's construction, leading to a conjectural discussion in Chapter 12 of how to compute μ₃ and μ₄ on Joyce's manifolds.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2023

Author dc:creator

• Mayther, Laurence

Advisor dc:contributor.advisor

• Kovalev, Alexei

• [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

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/34200843-fc05-419f-b794-cf8729b4484c/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.104430

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/361915