Positive mass theorem and the CR Yamabe equation on 5-dimensional contact spin manifolds

We consider the CR Yamabe equation with critical Sobolev exponent on a closed contact manifold M of dimension 2n+1. The problem of finding solutions with minimum energy has been resolved for all dimensions except for dimension 5 (n=2). In this paper we prove the existence of minimum energy solutions in the 5-dimensional case when M is spin. The proof is based on a positive mass theorem built up through a spinorial approach.