An explicit formula for the Webster torsion of a pseudo-hermitian manifold and its application to torsion-free hypersurfaces

Abstract

This paper gives an explicit formula for calculating the Webster pseudo torsion for a strictly pseudoconvex pseudo-hermitian hypersurface. As applications, we are able to classify some pseudo torsion-free hypersurfaces, which include real ellipsoids.

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

“…According to the formula given by Li and Luk [24] for the pseudotorsion, for any w ∈ H z (E(A)), one has that…”

Remarks on variational problems for Fefferman's measure

“…In [Ham] θ is chosen to be 2 −4/3 i(∂ρ − ∂ρ) and the Webster curvature is found to be − 2 5/3 3 κ in general and − 2 4/3 R 4/3 on the sphere of radius R. In [LiLu1] and [Li], on the other hand, θ is chosen to be − i 2 (∂ρ − ∂ρ) leading to curvature values which are −2 −1/3 times those in [Ham]; thus the Webster curvature is now 2 4/3 3 κ in general and 2 R 4/3 on the sphere of radius R. Also, with this choice of θ, the main formula in [LiLu2] can be used to check that the absolute value of the torsion coefficient is 2 2/3 γ. From Corollary B in [JL2] we have that (6.1) is minimized when g is constant; that is,…”

Webster pseudo-torsion formulas of CR manifolds

“…To solve the equivalence problem of pseudohermitian manifold, Webster derived the structure equations for M , from which the Webster Ricci curvature and Webster torsion tensor are defined. In [3], the author derived a formula for Webster pseudo-torsion for a real hypersurface in C n+1 . In this article, we derive a formula for Webster pseudo-torsion for the link of an isolated singularity of a n-dimensional complex subvariety in C n+1 and we present an alternative proof of the Li-Luk formula for Webster pseudo-torsion for a real hypersurface in C n+1 [3].…”

“…In [3], the author derived a formula for Webster pseudo-torsion for a real hypersurface in C n+1 . In this article, we derive a formula for Webster pseudo-torsion for the link of an isolated singularity of a n-dimensional complex subvariety in C n+1 and we present an alternative proof of the Li-Luk formula for Webster pseudo-torsion for a real hypersurface in C n+1 [3]. The main idea of the alternative proof is to describe the CR structure using all Euclidean coordinates z 1 , z 2 , .…”