On the Sobolev quotient of three-dimensional CR manifolds

We exhibit examples of compact three-dimensional CR manifolds of positive Webster class, Rossi spheres , for which the pseudo-hermitian mass, as defined by Cheng–Malchiodi–Yang (2017), 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.