arXiv subjects

Laurence H. Mayther

Publications and source records attributed to Laurence H. Mayther.

7 recordsLinked to original sources

arXiv2026

A general collapsing result for families of stratified Riemannian metrics on orbifolds

This paper proves a general collapsing result for families of stratified Riemannian metrics $\widehat{g}^μ$ on a compact orbifold $E$, subject to suitable limiting conditions on the metrics $\widehat{g}^μ$ as $μ\to \infty$. The result is distinct from similar theorems in the literature since it does not require bounds on curvature or injectivity radius of $\left(E,\widehat{g}^μ\right)$ and thus allows for Gromov-Hausdorff limits of $\left(E,\widehat{g}^μ\right)$ which have strictly lower dimension than $E$. The paper also introduces and studies a new class of stratified fibrations between orbifolds, termed weak submersions, and new classes of geometric structures on orbifolds, termed stratified Riemannian metrics, stratified Riemannian semi-metrics and stratified quasi-Finslerian structures, all of which play a key role in the proof of the main theorem.

math.DG↗

arXiv2026

Unboundedness above of the Hitchin functional on $\mathrm{G}_2$ 3-forms and associated collapsing results

This paper uses scaling arguments to prove the unboundedness above of the Hitchin functional on closed $\mathrm{G}_2$ 3-forms for two explicit closed 7-manifolds. The first manifold is the product $X \times S^1$ (where $X$ is the Nakamura manifold constructed by de Bartolomeis-Tomassini) equipped with a 4-dimensional family of closed $\mathrm{G}_2$ 3-forms and is inspired by a short paper of Fernández. The second is the manifold recently constructed by Fernández-Fino-Kovalev-Muñoz. In the latter example, careful resolution of singularities is required, in order to ensure that the rescaled forms are cohomologically constant. By combining suitable geometric estimates with a general collapsing theorem for orbifolds recently obtained by the author, explicit descriptions of the large volume limits of both manifolds are also obtained. The proofs in this paper are notable for not requiring explicit solution of the Laplacian flow evolution PDE for closed $\mathrm{G}_2$-structures, thereby allowing treatment of manifolds which lack the high degree of symmetry generally required for Laplacian flow to be explicitly soluble.

math.DG↗

arXiv2026

Unboundedness above and below of the Donaldson-Hitchin functionals on $\mathrm{G}_2$- and $\widetilde{\mathrm{G}}_2$-forms

This paper combines explicit local calculations with covering arguments to prove the unboundedness above and below (in a logarithmic sense) of the Donaldson-Hitchin functionals on $\mathrm{G}_2$ 4-forms, $\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms, over compact manifolds (or, more generally, orbifolds) with boundary. In addition, the Donaldson-Hitchin functional on $\mathrm{G}_2$ 3-forms over compact manifolds (or orbifolds) with boundary is shown to be unbounded below. As scholia, the critical points of the functionals on $\mathrm{G}_2$ 4-forms, $\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms are shown to be saddles, and initial conditions of the Laplacian coflow which do not lead to convergent solutions are shown to be dense.

math.DG↗

arXiv2026

Spectral invariants of Joyce orbifolds

This paper introduces two new spectral invariants of torsion-free $\mathrm{G}_2$-structures on closed orbifolds and computes their values on all Joyce orbifolds. These invariants are shown to be more discerning than the $\overlineν$-invariant of Crowley-Goette-Nordström when applied to Joyce orbifolds, and thus provide candidate tools for distinguishing between Joyce manifolds. The invariants may be viewed as regularisations of the classical Morse indices of the critical points of the Hitchin functionals on closed and coclosed $\mathrm{G}_2$-structures respectively. In the case of Joyce orbifolds, an interesting link with twisted Epstein $ζ$-functions is also observed.

math.DG↗

arXiv2026

Relative $h$-principles for closed stable forms

This paper uses convex integration to develop a new, general method for proving relative $h$-principles for closed, stable, exterior forms on manifolds. This method is applied to prove the relative $h$-principle for 4 classes of closed stable forms which were previously not known to satisfy the $h$-principle, $\textit{viz.}$ stable $(2k-2)$-forms in $2k$ dimensions, stable $(2k-1)$-forms in $2k+1$ dimensions, $\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms. The method is also used to produce new, unified proofs of all three previously established $h$-principles for closed, stable forms, $\textit{viz.}$ the $h$-principles for closed stable 2-forms in $2k+1$ dimensions, closed $\mathrm{G}_2$ 4-forms and closed $\mathrm{SL}(3;\mathbb{C})$ 3-forms. In addition, it is shown that if a class of closed stable forms satisfies the relative $h$-principle, then the corresponding Hitchin functional (whenever defined) is necessarily unbounded above. Due to the general nature of the $h$-principles considered in this paper, the application of convex integration requires an analogue of Hodge decomposition on arbitrary $n$-manifolds (possibly non-compact, or with boundary) which cannot, to the author's knowledge, be found elsewhere in the literature. Such a decomposition is proven in Appendix A.

math.DG↗

arXiv2026

The relative $h$-principle for closed $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms

This paper uses convex integration with avoidance and transversality arguments to prove the relative $h$-principle for closed $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms on oriented 6-manifolds. As corollaries, it is proven that if an oriented 6-manifold $\mathrm{M}$ admits any $\mathrm{SL}(3;\mathbb{R})^2$ 3-form, then every degree 3 cohomology class on $\mathrm{M}$ can be represented by an $\mathrm{SL}(3;\mathbb{R})^2$ 3-form and, moreover, that the corresponding Hitchin functional on $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms representing this class is necessarily unbounded above. Essential to the proof of the $h$-principle is a careful analysis of the rank 3 distributions induced by an $\mathrm{SL}(3;\mathbb{R})^2$ 3-form and their interaction with generic pairs of hyperplanes. The proof also introduces a new property of sets in affine space, termed macilence, as a method of verifying ampleness.

math.GT↗

arXiv2026

Topological properties of closed $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms on manifolds

This paper uses algebro-topological techniques such as characteristic classes and obstruction theory, together with the $h$-principles for $\widetilde{\mathrm{G}}_2$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms recently established by the author and the $h$-principle for $\mathrm{SL}(3;\mathbb{C})$ forms established by Donaldson, to prove results on the topological properties of closed $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms on oriented 6- and 7-manifolds. Specifically, a criterion for an arbitrary oriented 7-manifold to admit a closed (resp. coclosed) $\widetilde{\mathrm{G}}_2$-structure is obtained, proving a conjecture of Lê; a generalisation of Donaldson's '$\mathrm{G}_2$-cobordisms' to $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms is introduced, with homotopic $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms in a given cohomology class shown to be $\widetilde{\mathrm{G}}_2$-cobordant, a result which currently has no analogue in the $\mathrm{G}_2$ case; and a complete classification of closed $\mathrm{SL}(3;\mathbb{C})$ forms up to homotopy is provided. Additionally, a lower bound on the number of homotopy classes of closed $\mathrm{SL}(3;\mathbb{R})^2$ forms on a given manifold is obtained, and the question of which closed $\mathrm{SL}(3;\mathbb{C})$ or $\mathrm{SL}(3;\mathbb{R})^2$ forms arise as the boundary values of closed $\widetilde{\mathrm{G}}_2$-structures on oriented 7-manifolds is investigated.

math.AT↗