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

TL;DR

This paper proves that the Donaldson-Hitchin functionals on certain G2-forms are unbounded above and below, and analyzes their critical points and the behavior of the Laplacian coflow on compact manifolds and orbifolds.

Contribution

It establishes the unboundedness of the Donaldson-Hitchin functionals on G2-forms and characterizes their critical points as saddles, with implications for the Laplacian coflow.

Findings

Functionals are unbounded above and below on G2-forms.

Critical points are saddle points.

Non-convergent initial conditions are dense for the Laplacian coflow.

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.