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

Abstract

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 Fern\'{a}ndez. The second is the manifold recently constructed by Fern\'{a}ndez-Fino-Kovalev-Mu\~{n}oz. 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.