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We consider three-dimensional CR manifolds, which are modelled on the Heisenberg group.We introduce a natural concept of “mass” and prove its positivity under the condition that the scalar curvature is positive and in relation to their (holomorphic) embeddability properties.We apply this result to the CR Yamabe problem, and we discuss extremality of Sobolev-type quotients, giving some counterexamples for “Rossi spheres”.

This is joint work with J.H.Cheng and P.Yang.