In this paper the authors study the conformal geometry of spherical qc manifolds.
Recall that a spherical qc manifold is a locally conformally flat pseudo-Riemannian manifold admitting an atlas whose local charts are open subsets of the quaternionic Heisenberg group and whose transition functions take their values in the group \(\mathrm{Sp}(n+1,1)\). In this way they are the quaternionic counterparts of locally conformally flat Lorentzian manifolds (locally modelled on the Minkowski space with transition functions in \(\mathrm{SO}(n+1,1)\)) and spherical CR manifolds (locally modelled on the complex Heisenberg group with transition functions in \(\mathrm{SU}(n+1,1)\)).
The main results of the paper are as follows. The authors work out two constructions to obtain an abundance of spherical qc manifolds. The first one is based on connected sums (Section 2) and the second one is based on quotients of the standard spherical qc manifold by co-compact subgroups of \(\mathrm{Sp}(n+1,1)\) (Section 5). Then they investigate the conformal geometry of spherical qc manifolds (Sections 3 and 4) by introducing an appropriate Yamabe operator and its Green function. It is proved that for a connected compact spherical qc manifold exactly one of the three possibilities holds: it admits a conformally equivalent metric with everywhere positive, zero or negative scalar curvature. They accordingly call a spherical qc manifold scalar positive, zero or negative. It is proved that in the compact, scalar positive case, the Green function always exists and its regular part gives rise to a conformally invariant tensor (Theorem 4.1). A positive mass conjecture in this setting is formulated and it is proved that if it holds then this conformally invariant tensor itself gives rise to a spherical qc metric (Corollary 4.1). Finally in Section 6, they integrate the Green function against the so-called Patterson-Sullivan measure in order to construct a spherical qc metric of Nayatani-type (Theorem 6.1).