Embedded three-dimensional CR manifolds and the non-negativity of Paneitz operators

Source abstract

Let Ω ⊂ C 2 \Omega \subset \mathbb {C}^{2} be a strictly pseudoconvex domain and M = ∂ Ω M=\partial \Omega be a smooth, compact and connected CR manifold embedded in C 2 \mathbb {C}^2 with the CR structure induced from C 2 \mathbb {C}^{2} . The main result proved here is as follows. Assume the CR structure of M M has zero torsion. Then if we make a small real-analytic deformation of the CR structure of M M along embeddable directions, the CR structures along the deformation path continue to have non-negative Paneitz operators. We also show that any ellipsoid in C 2 \mathbb {C}^2 has positive Webster curvature.