On the CR analogue of Obata's theorem in a pseudohermitian 3-manifold
type: Item
instance of: scholarly article
title: On the CR analogue of Obata's theorem in a pseudohermitian 3-manifold
zbMATH Open document ID: 1182.32012
author: Shu-Cheng Chang
author: Hung-Lin Chiu
published in: Mathematische Annalen
publication date: 2009-08-27T00:00:00Z
full work available at URL: 04.pdf
review text: Let \((M, H^{10}, \theta)\) be a closed pseudo-Hermitian manifold of dimension \(2n+1\) that satisfies the following inequality for any vector field \(Z\) in the holomorphic distribution \(H^{10} \subset T^{\mathcal C} M\) \[ Ric_x(Z, Z) - \frac{n+1}{2} Tor_x(Z, Z) \geq k_0 d \vartheta(Z, \bar Z)\;,\tag{*} \] where \(Ric\) and \(Tor\) are the pseudo-Hermitian Ricci curvature and torsion, respectively, and \(k_0\) is some positive constant. Under some additional hypothesis, it has been proved that the first positive eigenvalue \(\mu_1\) of the sub-laplacian satisfies the inequality \(\mu_1 \geq \frac{n k_0}{n+1}\). Moreover, when \(M\) is the sphere \(S^{2n+1}\) with its standard pseudo-Hermitian structure, the equality holds. It is therefore conjectured the possibility of proving a CR analogue of Obata's theorem, namely that if \((M, H^{10}, \theta)\) satisfies (*) and \(\mu_1 = \frac{n k_0}{n+1}\), then \((M, H^{10}, \theta)\) is the standard pseudo-Hermitian sphere. In this paper, the authors prove an analogue of the Bochner formula on 3-dimensional CR manifolds, involving the CR Paneitz operators, and explore various of its consequences. In particular, for the pseudo-Hermitian closed 3-manifolds, they prove that the above conjecture is true when the torsion is vanishing and obtain a CR analogue of Li-Yau eigenvalue estimates for \(\mu_1\) when the torsion is non-vanishing and the CR Paneitz operator is nonnegative.
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 32V20
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 5598894
zbMATH Keywords: Obata's theorem
zbMATH Keywords: pseudohermitian 3-manifolds
reviewed by: Andrea Spiro
MaRDI profile type: Publication
OpenAlex ID: W2024074356
cites work: A fourth order curvature flow on a CR 3-manifold
cites work: Pseudo-Einstein and Q-flat metrics with eigenvalue estimates on CR-hypersurfaces
cites work: On the estimate of the first eigenvalue of a sub-Laplacian on a pseudo-Hermitian 3-manifold
cites work: The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold
cites work: Collapsing of Riemannian manifolds and eigenvalues of Laplace operator
cites work: Ambient metric construction of \(Q\)-curvature in conformal and CR geometries
cites work: CR invariant powers of the sub-Laplacian
cites work: Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains
cites work: The first eigenvalue of a sublaplacian on a pseudohermitian manifold
cites work: Q3135234
cites work: CR-structures on Seifert manifolds
cites work: Psuedo-Einstein Structures on CR Manifolds
cites work: The Fefferman Metric and Pseudohermitian Invariants
cites work: The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold
cites work: Certain conditions for a Riemannian manifold to be isometric with a sphere
DOI: 10.1007/S00208-009-0339-3
Recommended article: AN OBATA-TYPE THEOREM ON A THREE-DIMENSIONAL CR MANIFOLD
Recommended article: An Obata-type theorem in CR geometry
Recommended article: Q3626639
Recommended article: The diameter estimate and its application to CR Obata's theorem on closed pseudohermitian \((2n+1)\)-manifolds
Recommended article: On CR volume growth estimate in a complete pseudohermitian 3-manifold
Recommended article: Pseudo-conformal invariants of type (1,3) of CR manifolds
Recommended article: On Pseudohermitian Immersions Between Strictly Pseudoconvex CR Manifolds
Recommended article: A rigidity theorem for CR manifolds and a refinement of Obata and Lelong-Ferrand's result
Recommended article: Extension of \(CR\) structures on three dimensional compact pseudoconvex CR manifolds
Recommended article: On the pseudo-Hermitian sectional curvature of a strictly pseudoconvex CR manifold
description: scientific article; zbMATH DE number 5598894