Stable forms and special metrics

TL;DR

This work presents a unified variational framework based on GL(V)-invariant volume functionals on stable p-forms to produce and study special geometric structures in dimensions 6–8, including holonomy $G_2$, Spin($7$), and weak holonomy $SU(3)$ and PSU(3). By analyzing critical points of the volume on fixed cohomology classes and constrained variants, it links these forms to concrete geometric structures and PDEs, such as the Rarita–Schwinger equations, and shows how gradient and Hamiltonian flows generate Spin($7$) and $G_2$ metrics on product manifolds. The approach is practical for homogeneous and cohomogeneity-one constructions, yielding new explicit Spin($7$) and $G_2$ metrics, and it includes a detailed PSU(3) analysis in 8D with implications for Ricci components and invariant spinor fields. An appendix provides explicit volume definitions for all stability cases, enabling explicit computations in the described variational problems.

Abstract

We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general formalism becomes a practical tool for calculating homogeneous or cohomogeneity one examples, and we illustrate this with some newly discovered cases of Spin(7) and G2 metrics.

Theorems & Definitions (7)

• Theorem 1
• Lemma 2
• Theorem 3
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8