A positive mass theorem for CR manifolds

We consider a class of CR manifold which are defined as asymptotically

Heisenberg,

and for these we give a notion of mass. From the solvability of the

$\Box_b$ equation

in a certain functional class ([Hsiao-Yung]), we prove positivity of the

mass under the

condition that the Webster curvature is positive and that the manifold

is embeddable.

We apply this result to the Yamabe problem for compact CR manifolds,

assuming positivity

of the Webster class and non-negativity of the Paneitz operator. This is

joint work with

J.H.Cheng and P.Yang.