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On the Sobolev quotient in sub-Riemannian geometry. Malchiodi, Andrea


We consider a class of three-dimensional ``CR manifolds'' which are modelled on the Heisenberg group. We introduce a natural concept of ``mass'' and prove its positivity under the conditions that the Webster curvature is positive and in relation to their (holomorphic) embeddability properties. We apply this result to the CR Yamabe problem, and we discuss the properties of Sobolev-type quotients, giving some counterexamples to the existence of minimisers for ``Rossi spheres'', in sharp contrast to the Riemannian case. This is joint work with J.H.Cheng and P.Yang.

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