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On the Sobolev quotient of three-dimensional CR manifolds

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arxiv 1904.04665 v1 pith:DIOAP6MF submitted 2019-04-09 math.DG math.AP

Jih-Hsin Cheng , Andrea Malchiodi , Paul Yang This is my paper · ORCID
classification math.DGmath.AP
keywords manifoldsquotientthree-dimensionalarisesattainedcaseciteclass
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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.

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  1. Nonnegativity of the CR Paneitz operator for embeddable CR manifolds

    math.DG 2019-08 accept novelty 8.0 of 10

    For every closed embeddable strictly pseudoconvex CR three-manifold, the CR Paneitz operator is nonnegative and its kernel is exactly the CR pluriharmonic functions, giving a CR Yamabe solution for this class.

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