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 $G_{2}$ 4-forms, ${\tilde{G}}_{2}$ 3-forms and ${\tilde{G}}_{2}$ 4-forms, over compact manifolds (or, more generally, orbifolds) with boundary. In addition, the Donaldson-Hitchin functional on $G_{2}$ 3-forms over compact manifolds (or orbifolds) with boundary is shown to be unbounded below. As scholia, the critical points of the functionals on $G_{2}$ 4-forms, ${\tilde{G}}_{2}$ 3-forms and ${\tilde{G}}_{2}$ 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 $G_{2}$ - and ${\tilde{G}}_{2}$ -forms

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Unboundedness above and below of the Donaldson-Hitchin functionals on $G_{2}$ - and ${\tilde{G}}_{2}$ -forms

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Unboundedness above and below of the Donaldson-Hitchin functionals on G2G2​- and G~2G2​-forms

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Unboundedness above and below of the Donaldson-Hitchin functionals on G2G2​- and G~2G2​-forms

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Unboundedness above and below of the Donaldson-Hitchin functionals on $G_{2}$ - and ${\tilde{G}}_{2}$ -forms

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Unboundedness above and below of the Donaldson-Hitchin functionals on $G_{2}$ - and ${\tilde{G}}_{2}$ -forms

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