Solution of the qc Yamabe equation on a 3-Sasakian manifold and the quaternionic Heisenberg group

A complete solution to the quaternionic contact Yamabe equation on the qc sphere of dimension $4n+3$ as well as on the quaternionic Heisenberg group is given. A uniqueness theorem for the qc Yamabe problem in a compact locally 3-Sasakian manifold is shown.