Extremals for the Sobolev-Folland-Stein inequality, the quaternionic contact Yamabe problem and related geometric structures

We describe explicitly non-negative extremals for the Sobolev inequality on the quaternionic Heisenberg groups and determine the best constant in the $L^2$ Folland-Stein embedding theorem involving quaternionic contact (qc) geometry and the qc Yamabe equation. Translating the problem to the 3-Sasakian sphere, we determine the qc Yamabe invariant on the spheres. We describe explicitly all solutions to the qc Yamabe equation on the seven dimensional quaternionic Heisenberg group. The main tool is the notion of qc structure and the Biquard connection. We define a curvature-type tensor invariant called qc conformal curvature in terms of the curvature and torsion of the Biquard connection and show that a qc manifold is locally qc conformal (gauge equivalent) to the standard flat qc structure on the Heisenberg group, or equivalently, to the 3-sasakian sphere if and only if the qc conformal curvature vanishes. Possibly, this will help to reduce the qc Yamabe problem to that of the spherical qc manifolds.