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

Abstract

Let Ω ⊂ C 2 be a strictly pseudoconvex domain and M = ∂Ω be a smooth, compact and connected CR manifold embedded in C 2 with the CR structure induced from C 2 . The main result proved here is as follows. Assume the CR structure of M has zero torsion. Then if we make a small real-analytic deformation of the CR structure of M along embeddable directions, the CR structures along the deformation path continue to have nonnegative Paneitz operators. We also show that any ellipsoid in C 2 has positive Webster curvature. introductionThroughout this paper, we will use the notation and terminology in ([14]) unless otherwise specified. Let (M, J, θ) be a smooth, closed and connected three-dimensional pseudohermitian manifold, where θ is a contact form and J is a CR structure compatible with the contact bundle ξ = ker θ. The CR structure J decomposes C ⊗ξ into the direct sum of T 1,0 and T 0,1 which are eigenspaces of J with respect to i and −i, respectively. The Levi form , L θ is the Hermitian form on T 1,0 defined by Z, W L θ = −i dθ, Z ∧ W . We can extend , L θ to T 0,1 by defining Z, W L θ = Z, W L θ for all Z, W ∈ T 1,0 . The Levi form induces a natural Hermitian form on the dual bundle of T 1,0 , denoted by , L * θ , and hence on all the induced tensor bundles. Integrating the hermitian form (when acting on sections) over M with respect to the volume form dV = θ ∧ dθ, we get an inner product on the space of sections of each tensor bundle. We denote the inner product by the notation , . For2000 Mathematics Subject Classification. Primary 32V05, 32V20; Secondary 53C56. v19.

Plurisubharmonicity for the solution of the Fefferman equation and applications

“…It was proved by the Li [14] that ρ is strictly plurisubharmonic when D is a bounded domain in C n whose boundary is a real ellipsoid. In particular, when n = 2 case, this result was also proved by Chanillo, Chiu and Yang [2] later.…”

The CR Paneitz operator and the stability of CR pluriharmonic functions

“…The assumption that J t varies real-analytically in t is only utilized to show that S is open. It enters only because we use [3,Theorem 1.7].…”

“…As a partial converse, one would like to know if CR manifolds embedded in C 2 with some additional nice properties satisfy these nonnegativity conditions. Working in this direction, Chiu and the second two authors showed [3] that these nonnegative conditions hold for small deformations of a strictly pseudoconvex hypersurface with vanishing torsion in C 2 .…”

“…Since the standard contact form on the CR three-sphere is torsionfree, its CR Paneitz operator is nonnegative and has kernel consisting only of the CR pluriharmonic functions [7]. An elementary calculation [3] shows that the real ellipsoids have positive Webster scalar curvature, and thus satisfy condition (4) of Theorem 1.3. The condition (5) then follows from the stability results of [2,13].…”

A remark on the kernel of the CR Paneitz operator

“…(3) For the structure J 0 we have P 4 ψ, ψ ≥ 0 for all functions ψ and ker P 0 4 = P 0 , the space of CR pluriharmonic functions with respect to J 0 . In view of [4] the hypotheses of theorem (1.2) are satisfied by the family of ellipsoids in C 2 . Theorem 1.1 shows that the stability of the CR pluriharmonic functions plays a role in preventing the existence of the supplementary space.…”