  We exhibit examples of compact three-dimensional CR manifolds of positive
Webster class, {\em Rossi spheres}, for which the pseudo-hermitian mass as
defined in \cite{CMY17} is negative, and for which the infimum of the
CR-Sobolev quotient is not attained. To our knowledge, this is the first
geometric context on smooth closed manifolds where this phenomenon arises, in
striking contrast to the Riemannian case.

Differential Geometry;Analysis of PDEs