The Yamabe operator and invariants on octonionic contact manifolds and convex cocompact subgroups of ${\rm F}_{4(-20)}$

TL;DR

This paper develops the theory of the Yamabe operator on octonionic contact manifolds, explores their scalar invariants, and connects geometric properties with group actions of convex cocompact subgroups of F4(-20).

Contribution

It constructs the OC Yamabe operator, establishes its transformation properties, and links scalar positivity to the Poincaré exponent of certain subgroups, extending geometric analysis in octonionic settings.

Findings

The OC Yamabe operator's transformation formula under conformal changes.

Scalar positivity of OC manifolds correlates with the Yamabe invariant.

Connection between the Poincaré exponent and scalar positivity for subgroups of F4(-20).

Abstract

An octonionic contact (OC) manifold is always spherical. We construct the OC Yamabe operator on an OC manifold and prove its transformation formula under conformal OC transformations. An OC manifold is scalar positive, negative or vanishing if and only if its OC Yamabe invariant is positive, negative or zero, respectively. On a scalar positive OC manifold, we can construct the Green function of the OC Yamabe operator, and apply it to construct a conformally invariant tensor. It becomes an OC metric if the OC positive mass conjecture is true. We also show the connected sum of two scalar positive OC manifolds to be scalar positive if the neck is sufficiently long. On the OC manifold constructed from a convex cocompact subgroup of ${\rm F}_{4(-20)}$, we construct a Nayatani type Carnot-Carath\'eodory metric. As a corollary, such an OC manifold is scalar positive, negative or vanishing if and only if the Poincar\'e critical exponent of the subgroup is less than, greater than or equal to $10,$ respectively.