CR-structures on Seifert manifolds

In this paper, we first prove the CR analogue of Obata's theorem on a closed pseudohermitian 3-manifold with zero pseudohermitian torsion. Secondly, instead of zero torsion, we have the CR analogue of Li-Yau's eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub-Laplacian in a closed pseudohermitian 3-manifold with nonnegative CR Paneitz operator P 0 . Finally, we have a criterion for the positivity of first eigenvalue of the sub-Laplacian on a complete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator. The key step is a discovery of integral CR analogue of Bochner formula which involving the CR Paneitz operator.

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“…In additional, if M is simply connected, then (M, J, θ) is the standard pseudohermitian 3-sphere. We refer to [12] for some details.…”

Section: The Proofsmentioning

confidence: 99%

On the CR analogue of Obata’s theorem in a pseudohermitian 3-manifold

Chang

Chiu

2009

Math. Ann.

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“…There have been many studies in various aspects for this structure (e.g., [3], [18], [11], [14], [9], [24]). In this paper, we study the uniformization problem.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Uniformization of spherical CR manifolds

Chêng

Chiu

Yang

2014

Advances in Mathematics

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Abstract. Let M be a closed (compact with no boundary) spherical CR manifold of dimension 2n + 1. Let M be the universal covering of M. Let Φ denote a CR developing mapwhere S 2n+1 is the standard unit sphere in complex n + 1-space C n+1 . Suppose that the CR Yamabe invariant of M is positive. Then we show that Φ is injective for n ≥ 3. In the case n = 2, we also show that Φ is injective under the condition: s(M ) < 1. It then follows that M is uniformizable.

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“…Remark. All normal CR compact 3-manifolds are covered by circle bundles over a Riemann surface [7], [8], [1], see also next Section, and, if this circle bundle is not the Hopf fibration S 3 → S 2 , then all Sasakian structures are, up to a finite quotient, regular ( [1], see also next Section). If M is covered by S 3 , then any Sasakian structure on M is a deformation of a regular one [1], see next Section.…”

Section: Sasakian Geometrymentioning

confidence: 99%

“…it is the quotient of C 2 0 by a group G, generated by a normal finite subgroup H and by a holomorphic contraction g of C 2 as in the Proposition, in general with α, β ∈ C, 0 < |α| ≤ |β| < 1). We first need the orbits of V to be closed: as V := log αx∂ x + log βy∂ y , see [1], Proposition 8, (8) we obtain that α = |α|ε 1 and β = |β|ε 2 , where ε 1 , ε 2 are both primary n-roots of unity for n ∈ N * , i.e. ε k j = 1 ⇔ k = np, p ∈ Z, for j = 1, 2.…”

Section: Sasakian Geometrymentioning

confidence: 99%

“…Topologically, every compact normal CR (or Sasakian) 3-manifold is a Seifert fibration (Proposition 5, see also [8], [7] and [1]), but it turns out that the Sasakian structures themselves can be explicitly described on these manifolds: Theorem 1, Proposition 5 (these are extended versions of the results announced in [1]); In particular, if a compact Sasakian 3-manifold is not covered by S 3 (or non-spherical), its Sasakian structure is always regular, i.e. it is a (finite quotient of a) circle bundle over a Riemann surface and its metric arises from a Kaluza-Klein construction (Corollary 2), which leads to an elementary description of any Sasakian metric on these manifolds, either directly, or as deformations of a CR flat one.…”

Section: Introductionmentioning

confidence: 99%

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Normal CR structures on compact 3-manifolds

Belgun¹

2001

Mathematische Zeitschrift

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Abstract. We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotiens of S 3 or of a circle bundle over a Riemann surface of positive genus. In the latter case, we prove that their CR automorphisms group is a finite extension of S 1 , and we classify the normal CR structures on these manifolds. IntroductionAnalogs of complex manifolds in odd dimensions, pseudo-conformal CR manifolds are particular contact manifolds, with a complex structure on the corresponding distribution of hyperplanes, satisfying an integrability condition (see Section 2). Contrary to complex geometry, CR geometry is locally determined by a finite system of local invariants (like in the cases of conformal or projective structures), [17], [9], [16]. Therefore the space of locally non-isomorphic CR structures is a space with infinitely many parameters.In this paper, we focus our attention on normal CR manifolds, which admit global Reeb vector fields preserving the CR structure, in particular their CR automorphisms group has dimension at least 1. Our main result is that, for a compact normal CR 3-manifold, which is topologically not a quotient of S 3 , this CR automorphisms group is a finite extension of a circle, thus the Reeb vector field is unique up to a constant (Section 4, Theorem 2). This, together with the classification of Sasakian compact 3-manifolds (see Section 3), allows us to obtain the classification of normal CR structures on these manifolds (Section 4, Corollary 4).The question of classifying compact CR manifolds has first been solved in situations with a high order of local symmetry: the classification of flat compact CR manifolds, where the local CR automorphism group is P SU (n + 1, 1) (if the manifold has dimension 2n + 1), is due to E. Cartan [3] and to D. Burns and S. Shnider [2]; in dimension 3, homogeneous, simplyconnected, CR manifolds are either flat or (3-dimensional) Lie groups, and have been classified by E. Cartan [3] (see also [5]). In this case, there is no intermediate symmetry because E. Cartan has showed that a homogeneous CR manifold whose CR automorphism group has dimension greater than 3 is automatically flat.In dimension 3, the normal CR structures are always deformations of a flat one (Theorem 1, see also [1]), and the key point is that, for a CR Reeb vector field T , they admit compatible Sasakian metrics, for which T is 1 2 FLORIN ALEXANDRU BELGUN Killing (see Section 2 for details); these metrics are closely related to locally conformally Kähler metrics with parallel Lee form, natural analogs of Kähler structures on non-symplectic complex manifolds [19].Topologically, every compact normal CR (or Sasakian) 3-manifold is a Seifert fibration (Proposition 5, see also [8], [7] and [1]), but it turns out that the Sasakian structures themselves can be explicitly described on these manifolds: Theorem 1, Proposition 5 (these are ext...

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CR-structures on Seifert manifolds

Cited by 47 publications

References 37 publications

On the CR analogue of Obata’s theorem in a pseudohermitian 3-manifold

On the CR analogue of Obata’s theorem in a pseudohermitian 3-manifold

Uniformization of spherical CR manifolds

Normal CR structures on compact 3-manifolds

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