Reduction of Five-Dimensional Uniformly Levi Degenerate CR Structures to Absolute Parallelisms

Isaev, Alexander; Zaitsev, Dmitri

doi:10.1007/s12220-013-9419-4

Cited by 41 publications

(39 citation statements)

References 31 publications

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“…There exists a proper real-analytic subset V Ă M such that the complement M zV is either (i) Levi nondegenerate or (ii) uniformly Levi-degenerate of rank 1 and 2-nondegenerate (see [BER,§11.1] for the definition of k-nondegeneracy). In case (ii), by [IZ,Corollary 5.4] one has dim s ď 10. In case (i), if M is nonspherical at some p P M zV , by [Kr] we have dim s ď 8.…”

Section: 2mentioning

confidence: 94%

“…For n " 1 the above conjecture holds true since a 3-dimensional holomorphically nondegenerate CR-hypersurface always has points of Levi-nondegeneracy. For n " 2 the conjecture was established in [IZ,Corollary 5.8] where the proof relied on the reduction of 5-dimensional uniformly Levi-degenerate 2-nondegenerate CRstructures to absolute parallelisms. Thus, for real-analytic connected holomorphically nondegenerate CR-hypersurfaces of CR-dimension n " 1, 2 one has, just as in the Levi-nondegenerate case, D max " n 2`4 n`3.…”

Section: Introductionmentioning

confidence: 99%

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On the symmetry algebras of 5-dimensional CR-manifolds

Isaev

Kruglikov

2017

Advances in Mathematics

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Abstract. We show that for a real-analytic connected holomorphically nondegenerate 5-dimensional CR-hypersurface M and its symmetry algebra s one has either: (i) dim s " 15 and M is spherical (with Levi form of signature either p2, 0q or p1, 1q everywhere), or (ii) dim s ď 11 where dim s " 11 can only occur if on a dense open subset M is spherical with Levi form of signature p1, 1q. Furthermore, we construct a series of examples of pairwise nonequivalent CR-hypersurfaces with dim s " 11.

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Section: 2mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

On the symmetry algebras of 5-dimensional CR-manifolds

Isaev

Kruglikov

2017

Advances in Mathematics

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“…The rank one complex subbundle D 1,0 ⊂ C ⊗ T M is involutive since clearly [D 1,0 , D 1,0 ] ⊂ D 1,0 . Then by definition ( [15]), D 1 is a CR structure and the real submanifold M is a CR manifold of CR dimension 1 = 1 2 rank D 1 and codimension k = dim R M − 2 CRdim M . As before, we can denote D 1 , D 1,0 and D 0,1 by T c M , T 1,0 M and T 0,1 M , respectively.…”

Section: Introductionmentioning

confidence: 99%

Biholomorphic equivalence to totally nondegenerate model CR manifolds

Sabzevari

2018

Annali di Matematica

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ApplyingÉlie Cartan's classical method, we show that the biholomorphic equivalence problem to a totally nondegenerate Beloshapka's model of CR dimension one and codimension k > 1, whence of real dimension 2 + k, is reducible to some absolute parallelism, namely to an {e}-structure on a certain prolonged manifold of real dimension either 3 + k or 4 + k. The proof relies on the weight analysis of the structure equations associated with the mentioned problem of equivalence. Thanks to the achieved results, we prove Beloshapka's maximum conjecture about the rigidity of his CR models of certain lengths equal or greater than three: " CR automorphism Lie groups of these models do not contain any nonlinear map, preserving the origin ". Here, we mainly deal with CR models of the fixed CR dimension one though the results seem generalizable by means of certain analogous proofs.

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“…We emphasize that this is a different kind of connection than the one known in the literature (e.g. [7,31,9,14,28,18,22,4,6,30,16]). In contrast to situations considered in the cited work, where the connection arises due to certain "uniformity" of the CR-structure, in our case, already the rank of the Levi form is not constant.…”

Section: Canonical Connection Along the Levi Degeneracy Setmentioning

confidence: 86%

Convergent normal form and canonical connection for hypersurfaces of finite type in C2

Kossovskiy

Zaitsev

2015

Advances in Mathematics

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We study the holomorphic equivalence problem for finite type hypersurfaces in C 2 . We discover a geometric condition, which is sufficient for the existence of a natural convergent normal form for a finite type hypersurface. We also provide an explicit construction of such a normal form. As an application, we construct a canonical connection for a large class of finite type hypersurfaces. To the best of our knowledge, this gives the first construction of an invariant connection for Levi-degenerate hypersurfaces in C 2 .

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Reduction of Five-Dimensional Uniformly Levi Degenerate CR Structures to Absolute Parallelisms

Abstract: Let C 2,1 be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. We show that the CR-structures in C 2,1 are reducible to so(3, 2)-valued absolute parallelisms and give applications of this result.

Cited by 41 publications

References 31 publications

On the symmetry algebras of 5-dimensional CR-manifolds

On the symmetry algebras of 5-dimensional CR-manifolds

Biholomorphic equivalence to totally nondegenerate model CR manifolds

Convergent normal form and canonical connection for hypersurfaces of finite type in C2

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