An explicit formula for the Webster pseudo-Ricci curvature on real hypersurfaces and its application for characterizing balls in {$\Bbb C\sp n$}

Li, Song-Ying; Luk, Hing-Sun

doi:10.4310/cag.2006.v14.n4.a4

Cited by 19 publications

(13 citation statements)

References 26 publications

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“…We will need a formula for the Webster curvature for hypersurfaces embedded in C 2 in a form suitable for our computations. Other formulae have been derived in [16], see Theorem 1.1 there.…”

Section: The Webster Curvature For Ellipsoidsmentioning

confidence: 99%

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

Chanillo¹,

Chiu²,

Yang³

2013

Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations

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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.

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Section: The Webster Curvature For Ellipsoidsmentioning

confidence: 99%

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

Chanillo¹,

Chiu²,

Yang³

2013

Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations

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“…Proof. -This follows from (6.14) and a well-known formula for the Christoffel symbols of the Tanaka-Webster connection [17].…”

Section: The Ahlfors Tensor a In General Dimensionsmentioning

confidence: 76%

The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps

Lamel¹,

Son²

2022

Annales De l'Institut Fourier

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We study a CR analogue of the Ahlfors derivative for conformal immersions of Stowe that generalizes the CR Schwarzian derivative studied earlier by the second-named author. This notion possesses several important properties similar to those of the conformal counterpart and provides a new invariant for spherically equivalent CR maps from strictly pseudoconvex CR manifolds into a sphere. The invariant is computable and distinguishes many well-known sphere maps. In particular, it vanishes precisely when the map is spherically equivalent to the linear embedding of spheres.Résumé. -Nous étudions un analogue CR de la dérivée au sens d'Ahlfors pour les immersions conformes de Stowe, qui généralise la dérivée schwarzienne CR étudiée antérieurement par le second auteur. Cette notion possède plusieurs propriétés importantes similaires à celles de son homologue conforme et fournit un nouvel invariant pour les applications CR, sphériquement équivalentes, de variétés CR strictement pseudoconvexes à valeurs dans la sphère. Cet invariant est calculable et permet de distinguer beaucoup d'applications CR sphériques entre elles. En particulier, il s'annule précisément quand l'application est sphériquement équivalente au plongement linéaire entre sphères.

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“…According to the formula given by Li and Luk [23] for the pseudo-Ricci curvatures, for any w ∈ H z (E(A)), the Webster pseudo-Ricci curvature of (E(A), θ) is given by…”

Section: Proposition 21 Under the Assumption (15) One Hasmentioning

confidence: 99%

“…Therefore, the proof of Theorem 1.3 is complete. It is easy to see that on E 1 = {z ∈ E(A) : ρ n+1 (z) = 0}, by the setting in Li and Luk [23], we have…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

On the CR–Obata theorem and some extremal problems associated to pseudoscalar curvature on the real ellipsoids in $\mathbb{C}^{n+1}$

Li¹,

Tran²

2011

Trans. Amer. Math. Soc.

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An explicit formula for the Webster pseudo-Ricci curvature on real hypersurfaces and its application for characterizing balls in {$\Bbb C\sp n$}

Abstract: In the paper, we provide an explicit formula for computing the Webster pseudo Ricci curvature, we also apply this formula to obtain a theorem on characterizing balls by using area and pseudo scalar curvature.

Cited by 19 publications

References 26 publications

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

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

The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps

On the CR–Obata theorem and some extremal problems associated to pseudoscalar curvature on the real ellipsoids in $\mathbb{C}^{n+1}$

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