Nonnegativity of the CR Paneitz operator for embeddable CR manifolds

不可嵌入环面与CR Paneitz算子

Non-embeddable torus and CR Paneitz operator

研究不可嵌入三维环面上的CR Paneitz算子，在温和假设下证明其具有无穷多个负特征值，并给出具体例子。

非可嵌入CR流形上的CR Paneitz算子

CR Paneitz operator on non-embeddable CR manifolds

针对CR Paneitz算子在非可嵌入CR流形上的性质展开研究，证明其本质自伴、非零谱离散等特性，并通过Rossi球面例子揭示其有无穷多负特征值，与可嵌入情况形成显著差异。

Embedded Three Dimensional CR Manifolds and the Non-Negativity of Paneitz Operators

Let $Ω$ be a bounded strictly pseudoconvex domain in $C^2$ with a smooth, connected and compact boundary M and having a CR structure $J_0$ induced from $C^2$. Assume this CR structure has zero Webster torsion. Then if we deform the CR structure through real-analytic dependence on the deformation parameter and such that each deformed structure along the deformation path is smooth and embeddable in $C^2$, we show that for small deformations of the CR structure $J$ from $J_0$, the associated CR Paneitz operator for $J$ is non-negative. We also show that the Webster curvature for any ellipsoid in …

Pseudo-Einstein structure, eigenvalue estimate for the CR Paneitz operator and its applications to uniformization theorem

In this note, we mainly focus on the existence of pseudo-Einstein contact forms, an upper bound eigenvalue estimate for the CR Paneitz operator and its applications to the uniformization theorem for Sasakian space form in an embeddable closed strictly pseudoconvex CR 3-manifold. Firstly, the existence of pseudo-Einstein contact form is confirmed if the CR 3-manifold is Sasakian. Secondly, we derive an eigenvalue upper bound estimate for the CR Paneitz operator and obtain the CR uniformization theorem for a class of CR 3-manifolds. At the end, under the positivity assumption of the pseudohermit…

Embeddability for Three-Dimensional Cauchy-Riemann Manifolds and CR Yamabe Invariants

Let M^3 be a closed CR 3-manifold. In this paper we derive a Bochner formula for the Kohn Laplacian in which the pseudo-hermitian torsion plays no role. By means of this formula we show that the non-zero eigenvalues of the Kohn Laplacian are bounded below by a positive constant provided the CR Paneitz operator is non-negative and the Webster curvature is positive. Our lower bound for the non-zero eigenvalues is sharp and is attained on S^3. A consequence of our lower bound is that all compact CR 3-manifolds with non-negative CR Paneitz operator and positive CR Yamabe constant are embeddable. N…

Pseudo-Einstein and Q-flat metrics with eigenvalue estimates on CR-hypersurfaces

Let $M^{2n-1}$ be the smooth boundary of a bounded strongly pseudo-convex domain $Ω$ in a complete Stein manifold $V^{2n}$. Then (1) For $n \ge 3$, $M^{2n-1}$ admits a pseudo-Eistein metric; (2) For $n \ge 2$, $M^{2n-1}$ admits a Fefferman metric of zero CR Q-curvature; and (3) for a compact strictly pseudoconvex CR embeddable 3-manifold $M^3$, its CR Paneitz operator $P$ is a closed operator.