Cayley-positive cones and a Calabi-type conjecture for Spin(7)-structures

Determine further properties of the Cayley-positive cone \(C^{\Phi}(M)\) and the strict Cayley-positive cone \(K^{\Phi}(M)\) for compact Spin(7)-manifolds with torsion-free or arbitrary Spin(7)-structures, and develop an analogue of the Spin(7)-potential result that can formulate a Calabi-type conjecture in Spin(7)-geometry.

Background

The paper defines Cayley-positive and strictly Cayley-positive cones using closed representatives in the Bott–Chern-type cohomology of a compact Spin(7)-manifold. For torsion-free structures, it proves that these cones have basic convexity properties and that the Spin(7)-structure itself determines a class in the strict cone.

The paper also proves a potential theorem: two closed $4$-forms of the same de Rham class and lying in $\Omega^4_1\oplus\Omega^4_{35}$ differ by a $dd^{\Phi}$-potential, uniquely up to constants. The authors explicitly leave open the extension and deeper characterization of these cones, including a possible Calabi-type existence theory, for torsion-free and arbitrary Spin(7)-structures.