A positive mass theorem in three dimensional Cauchy-Riemann geometry

We define an ADM-like mass, called p-mass, for an asymptotically flat pseudohermitian manifold. The p-mass for the blow-up of a compact pseudohermitian manifold (with no boundary) is identified with the first nontrivial coefficient in the expansion of the G reen function for the CR Laplacian. We deduce an integral formula for the p-mass, and we reduce its po sitivity to a solution of Kohn’s equation. We prove that the p-mass is non-negative for (blow-ups of) compact 3-manifolds of positive Tanaka-Webster class and with non-negative CR Paneitz operator . Under these assumptions, we also characterize the zero mass case as the standard three dimension al CR sphere. We then show the existence of (non-embeddable) CR 3-manifolds having nonpositive Paneitz ope rator or negative p-mass through a second variation formula. Finally, we apply our main result to find solut ions of the CR Yamabe problem with minimal energy