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.