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

Laurence H. Mayther

TL;DR

This work studies the unboundedness properties of Donaldson–Hitchin functionals associated with $G_2$ and $\widetilde{G}_2$-structures on compact oriented $7$-manifolds (or orbifolds) with boundary. By combining explicit local computations with covering arguments, it shows that $\mathcal{H}_4^{\partial}$, $\widetilde{\mathcal{H}}_3^{\partial}$, and $\widetilde{\mathcal{H}}_4^{\partial}$ are unbounded above and below in a logarithmic sense on $[\psi]_{+}^{\partial}$ (for closed $G_2$ $4$-forms $\psi$), while $\mathcal{H}_3^{\partial}$ is unbounded below on $[\phi]_{+}^{\partial}$. The critical points of these functionals are saddles, and the Laplacian coflow from generic boundary-adhering initial data does not generically converge to torsion-free structures, with the non-convergent set dense in $C^0$. These results remain valid in the orbifold setting, and they yield a partial analogue for $\mathcal{H}_3^{\partial}$ regarding the saddle/flatness structure and convergence behavior. The paper also clarifies the relation between unboundedness and the long-time behavior of the Laplacian coflow, and highlights open questions for the unboundedness of $\mathcal{H}_3^{\partial}$ in full generality.

Abstract

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.

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

TL;DR

This work studies the unboundedness properties of Donaldson–Hitchin functionals associated with and -structures on compact oriented -manifolds (or orbifolds) with boundary. By combining explicit local computations with covering arguments, it shows that , , and are unbounded above and below in a logarithmic sense on (for closed -forms ), while is unbounded below on . The critical points of these functionals are saddles, and the Laplacian coflow from generic boundary-adhering initial data does not generically converge to torsion-free structures, with the non-convergent set dense in . These results remain valid in the orbifold setting, and they yield a partial analogue for regarding the saddle/flatness structure and convergence behavior. The paper also clarifies the relation between unboundedness and the long-time behavior of the Laplacian coflow, and highlights open questions for the unboundedness of in full generality.

Abstract

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 4-forms, 3-forms and 4-forms, over compact manifolds (or, more generally, orbifolds) with boundary. In addition, the Donaldson-Hitchin functional on 3-forms over compact manifolds (or orbifolds) with boundary is shown to be unbounded below. As scholia, the critical points of the functionals on 4-forms, 3-forms and 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.
Paper Structure (3 sections, 4 theorems, 11 equations)

This paper contains 3 sections, 4 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Orbifolds

Key Result

Theorem 1.1

The functionals $\mathcal{H}_{4}^\partial$, $\widetilde{\mathcal{H}}_{3}^\partial$ and $\widetilde{\mathcal{H}}_{4}^\partial$ are always unbounded above and below. Specifically, let $\mathrm{M}$ be a compact, oriented 7-manifold (or, more generally, 7-orbifold) with boundary and suppose that $\mathr Likewise, the analogous statement holds for the functionals $\widetilde{\mathcal{H}}_{3}^\partial$

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1