{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/361915"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/361915","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Hitchin Functionals, h-Principles and Spectral Invariants","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 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 $\\widetilde{\\mathrm{G}}_2$, $\\mathrm{SL}(3;\\mathbb{C})$ and $\\mathrm{SL}(3;\\mathbb{R})^2$ forms. Firstly, it characterises which oriented 7-manifolds admit closed $\\widetilde{\\mathrm{G}}_2$ forms, in the process introducing a new technique for proving the vanishing of natural cohomology classes on non-closed manifolds. Next, it introduces $\\widetilde{\\mathrm{G}}_2$-cobordisms of closed $\\mathrm{SL}(3;\\mathbb{C})$ and $\\mathrm{SL}(3;\\mathbb{R})^2$ 3-forms and proves that homotopic forms are $\\widetilde{\\mathrm{G}}_2$-cobordant. Additionally, Chapter 9 classifies $\\mathrm{SL}(3;\\mathbb{C})$ 3-forms up to homotopy and provides a partial classification result on homotopy classes of $\\mathrm{SL}(3;\\mathbb{R})^2$ 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 $\\mathrm{G}_2$-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 $\\mathrm{G}_2$-structures: the $\\overline{\\nu}$-invariant of Crowley-Goette-Nordström. Part III defines two new invariants of torsion-free $\\mathrm{G}_2$-structures, termed $\\mu_3$- and $\\mu_4$-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 $\\mathrm{G}_2$-manifolds constructed via Joyce's `generalised Kummer construction'. Chapter 11 obtains closed formulae for $\\mu_3$ and $\\mu_4$ on the orbifolds used in Joyce's construction, leading to a conjectural discussion in Chapter 12 of how to compute $\\mu_3$ and $\\mu_4$ on Joyce's manifolds.","abstract_html":"This thesis investigates Hitchin functionals and $h$-principles for stable forms on oriented manifolds, with a special focus on <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">G</span><sub>2</sub></span> and <span class=\"etd-inline-math\">\\widetilde{<span class=\"etd-inline-math-roman\">G</span>}<sub>2</sub></span> 3- and 4-forms. Additionally, it introduces two new spectral invariants of torsion-free <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">G</span><sub>2</sub></span>-structures. Part I begins by investigating an open problem posed by Bryant, <span class=\"etd-inline-math\"><em>viz.</em></span> whether the Hitchin functional <span class=\"etd-inline-math\">\\mathcal{H}<sub>3</sub></span> on closed <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">G</span><sub>2</sub></span> 3-forms is unbounded above. Chapter 3 uses a scaling argument to obtain sufficient conditions for the functional <span class=\"etd-inline-math\">\\mathcal{H}<sub>3</sub></span> to be unbounded above and applies this result to prove the unboundedness above of <span class=\"etd-inline-math\">\\mathcal{H}<sub>3</sub></span> on two explicit examples of closed 7-manifolds with closed <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">G</span><sub>2</sub></span> 3-forms. Chapter 3 then proceeds to interpret this unboundedness geometrically, demonstrating an unexpected link between the functional <span class=\"etd-inline-math\">\\mathcal{H}<sub>3</sub></span> and fibrations, proving that the &#x27;large volume limit&#x27; of <span class=\"etd-inline-math\">\\mathcal{H}<sub>3</sub></span> 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 <span class=\"etd-inline-math\">\\mathcal{H}<sub>4</sub></span>, <span class=\"etd-inline-math\">\\widetilde{\\mathcal{H}}<sub>3</sub></span> and <span class=\"etd-inline-math\">\\widetilde{\\mathcal{H}}<sub>4</sub></span> on closed <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">G</span><sub>2</sub></span> 4-forms, <span class=\"etd-inline-math\">\\widetilde{<span class=\"etd-inline-math-roman\">G</span>}<sub>2</sub></span> 3-forms and <span class=\"etd-inline-math\">\\widetilde{<span class=\"etd-inline-math-roman\">G</span>}<sub>2</sub></span> 4-forms respectively. In its main result, Chapter 5 proves that <span class=\"etd-inline-math\">\\mathcal{H}<sub>4</sub>,\\widetilde{\\mathcal{H}}<sub>3</sub>,\\widetilde{\\mathcal{H}}<sub>4</sub></span> are always unbounded above and below (whenever defined), and also that <span class=\"etd-inline-math\">\\mathcal{H}<sub>3</sub></span> is always unbounded below (whenever defined). As scholia, the critical points of the functionals <span class=\"etd-inline-math\">\\mathcal{H}<sub>4</sub></span>, <span class=\"etd-inline-math\">\\widetilde{\\mathcal{H}}<sub>3</sub></span> and <span class=\"etd-inline-math\">\\widetilde{\\mathcal{H}}<sub>4</sub></span> 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 <span class=\"etd-inline-math\">\\widetilde{<span class=\"etd-inline-math-roman\">G</span>}<sub>2</sub></span>, <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">SL</span>(3;\\mathbb{C})</span> and <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">SL</span>(3;\\mathbb{R})<sup>2</sup></span> forms. Firstly, it characterises which oriented 7-manifolds admit closed <span class=\"etd-inline-math\">\\widetilde{<span class=\"etd-inline-math-roman\">G</span>}<sub>2</sub></span> forms, in the process introducing a new technique for proving the vanishing of natural cohomology classes on non-closed manifolds. Next, it introduces <span class=\"etd-inline-math\">\\widetilde{<span class=\"etd-inline-math-roman\">G</span>}<sub>2</sub></span>-cobordisms of closed <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">SL</span>(3;\\mathbb{C})</span> and <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">SL</span>(3;\\mathbb{R})<sup>2</sup></span> 3-forms and proves that homotopic forms are <span class=\"etd-inline-math\">\\widetilde{<span class=\"etd-inline-math-roman\">G</span>}<sub>2</sub></span>-cobordant. Additionally, Chapter 9 classifies <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">SL</span>(3;\\mathbb{C})</span> 3-forms up to homotopy and provides a partial classification result on homotopy classes of <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">SL</span>(3;\\mathbb{R})<sup>2</sup></span> 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 <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">G</span><sub>2</sub></span>-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 <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">G</span><sub>2</sub></span>-structures: the $\\overline{\\nu}$-invariant of Crowley-Goette-Nordström. Part III defines two new invariants of torsion-free <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">G</span><sub>2</sub></span>-structures, termed <span class=\"etd-inline-math\">&mu;<sub>3</sub></span>- and <span class=\"etd-inline-math\">&mu;<sub>4</sub></span>-invariants, by regularising the classical notion of Morse index for the Hitchin functionals <span class=\"etd-inline-math\">\\mathcal{H}<sub>3</sub></span> and <span class=\"etd-inline-math\">\\mathcal{H}<sub>4</sub></span> at their critical points. In general, there is no known way to compute $\\overline{\\nu}$ for <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">G</span><sub>2</sub></span>-manifolds constructed via Joyce&#x27;s `generalised Kummer construction&#x27;. Chapter 11 obtains closed formulae for <span class=\"etd-inline-math\">&mu;<sub>3</sub></span> and <span class=\"etd-inline-math\">&mu;<sub>4</sub></span> on the orbifolds used in Joyce&#x27;s construction, leading to a conjectural discussion in Chapter 12 of how to compute <span class=\"etd-inline-math\">&mu;<sub>3</sub></span> and <span class=\"etd-inline-math\">&mu;<sub>4</sub></span> on Joyce&#x27;s manifolds.","abstract_has_math":true,"creators":["Mayther, Laurence"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Kovalev, Alexei"],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-06-30","date_published":"2023-06-30","updated_at":"2026-07-22T22:24:23Z","subjects":["[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"],"languages":["eng"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/34200843-fc05-419f-b794-cf8729b4484c/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.104430","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Kovalev, Alexei"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["Engineering and Physical Sciences Research Council Studentship 2261110"]},{"key":"dc:creator","label":"Author","values":["Mayther, Laurence"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2023-06-30"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/361915"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["[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"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/34200843-fc05-419f-b794-cf8729b4484c/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.17863/CAM.104430"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/e1715a54-5f4f-474b-a655-efd74b855d59/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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 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 $\\widetilde{\\mathrm{G}}_2$, $\\mathrm{SL}(3;\\mathbb{C})$ and $\\mathrm{SL}(3;\\mathbb{R})^2$ forms. Firstly, it characterises which oriented 7-manifolds admit closed $\\widetilde{\\mathrm{G}}_2$ forms, in the process introducing a new technique for proving the vanishing of natural cohomology classes on non-closed manifolds. Next, it introduces $\\widetilde{\\mathrm{G}}_2$-cobordisms of closed $\\mathrm{SL}(3;\\mathbb{C})$ and $\\mathrm{SL}(3;\\mathbb{R})^2$ 3-forms and proves that homotopic forms are $\\widetilde{\\mathrm{G}}_2$-cobordant. Additionally, Chapter 9 classifies $\\mathrm{SL}(3;\\mathbb{C})$ 3-forms up to homotopy and provides a partial classification result on homotopy classes of $\\mathrm{SL}(3;\\mathbb{R})^2$ 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 $\\mathrm{G}_2$-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 $\\mathrm{G}_2$-structures: the $\\overline{\\nu}$-invariant of Crowley-Goette-Nordström. Part III defines two new invariants of torsion-free $\\mathrm{G}_2$-structures, termed $\\mu_3$- and $\\mu_4$-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 $\\mathrm{G}_2$-manifolds constructed via Joyce's `generalised Kummer construction'. Chapter 11 obtains closed formulae for $\\mu_3$ and $\\mu_4$ on the orbifolds used in Joyce's construction, leading to a conjectural discussion in Chapter 12 of how to compute $\\mu_3$ and $\\mu_4$ on Joyce's manifolds."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["87eda9de84448d1f82354d60eee3eb5f","44c412eb2200c87cb392b3e1a26fd303"]},{"key":"dc:title","label":"Title","values":["Hitchin Functionals, h-Principles and Spectral Invariants"]}]}],"canonical_facts":{"dc:contributor.advisor":["Kovalev, Alexei"],"dc:contributor.sponsor":["Engineering and Physical Sciences Research Council Studentship 2261110"],"dc:creator":["Mayther, Laurence"],"dc:date.issued":["2023-06-30"],"dc:description.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 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 $\\widetilde{\\mathrm{G}}_2$, $\\mathrm{SL}(3;\\mathbb{C})$ and $\\mathrm{SL}(3;\\mathbb{R})^2$ forms. Firstly, it characterises which oriented 7-manifolds admit closed $\\widetilde{\\mathrm{G}}_2$ forms, in the process introducing a new technique for proving the vanishing of natural cohomology classes on non-closed manifolds. Next, it introduces $\\widetilde{\\mathrm{G}}_2$-cobordisms of closed $\\mathrm{SL}(3;\\mathbb{C})$ and $\\mathrm{SL}(3;\\mathbb{R})^2$ 3-forms and proves that homotopic forms are $\\widetilde{\\mathrm{G}}_2$-cobordant. Additionally, Chapter 9 classifies $\\mathrm{SL}(3;\\mathbb{C})$ 3-forms up to homotopy and provides a partial classification result on homotopy classes of $\\mathrm{SL}(3;\\mathbb{R})^2$ 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 $\\mathrm{G}_2$-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 $\\mathrm{G}_2$-structures: the $\\overline{\\nu}$-invariant of Crowley-Goette-Nordström. Part III defines two new invariants of torsion-free $\\mathrm{G}_2$-structures, termed $\\mu_3$- and $\\mu_4$-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 $\\mathrm{G}_2$-manifolds constructed via Joyce's `generalised Kummer construction'. Chapter 11 obtains closed formulae for $\\mu_3$ and $\\mu_4$ on the orbifolds used in Joyce's construction, leading to a conjectural discussion in Chapter 12 of how to compute $\\mu_3$ and $\\mu_4$ on Joyce's manifolds."],"dc:format.checksum.md5":["87eda9de84448d1f82354d60eee3eb5f","44c412eb2200c87cb392b3e1a26fd303"],"dc:identifier.doi":["https://doi.org/10.17863/CAM.104430"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/e1715a54-5f4f-474b-a655-efd74b855d59/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/361915"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/34200843-fc05-419f-b794-cf8729b4484c/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["[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"],"dc:title":["Hitchin Functionals, h-Principles and Spectral Invariants"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:23Z"}