Projective geometry of 3-Sasaki structures

We show that $3$-Sasaki structures admit a natural description in terms of projective differential geometry. This description provides a concrete link between $3$-Sasaki structures and several other geometries and constructions via a single unifying picture. First we establish that a $3$-Sasaki structure may be understood as a projective structure equipped with a certain holonomy reduction to the (possibly indefinite) unitary quaternionic group $\textrm{Sp}(p,q)$, namely a parallel hyperk\"ahler structure on the projective tractor bundle satisfying a particular genericity condition. For the converse, where one begins with a general parallel hyperk\"ahler structure on the projective tractor bundle, the genericity condition is not automatic. Indeed we prove that generically such a reduction decomposes the underlying manifold into a disjoint union of strata including open manifolds with (indefinite) $3$-Sasaki structures and a closed separating hypersurface at infinity with respect to the $3$-Sasaki metrics. Moreover, it is shown that the latter hypersurface inherits a Biquard-Fefferman conformal structure, which thus (locally) fibres over a quaternionic contact structure, and which in turn compactifies the natural quaternionic K\"ahler quotients of the $3$-Sasaki structures on the open manifolds. As an application we describe the projective compactification of (suitably) complete, non-compact (indefinite) $3$-Sasaki manifolds and recover Biquard's notion of asymptotically hyperbolic quaternionic K\"ahler metrics.