On the existence of the Yamabe problem on contact Riemannian manifolds

It was proved in [25] that for a contact Riemannian manifold with non-integrable almost complex structure, the Yamabe problem is subcritical in the sense that its Yamabe invariant is less than that of the Heisenberg group. In this paper we give a complete proof of the solvability of the contact Riemannian Yamabe problem in the subcritical case. These two results implies that the Yamabe problem on a contact Riemannian manifold is always solvable. By constructing normal coordinates on a contact Riemannian manifold, we can osculate the contact Riemannian structure at each point by the standard structure on the Heisenberg group. This osculation makes the machine of singular integral operators work on general contact Riemannian manifolds. We apply it to obtain the regularity of the SubLaplacians and the Yamabe equation, which allow us to solve the contact Riemannian Yamabe problem in the subcritical case by Jerison-Lee’s approach in the CR case. We also clarify two claims in their proof. M.S.C. 2010: 53D10, 53D35, 32V99.