On the symmetry algebras of 5-dimensional CR-manifolds

Isaev, Alexander; Kruglikov, Boris

doi:10.1016/j.aim.2017.10.020

Cited by 11 publications

(16 citation statements)

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“…This result improves on the statement of Conjecture 1.1. Note that the result is global in M, even if one takes M " U to be a small fixed neighborhood of a point x P M. The proof of the theorem also leads to the following local version of the result, generalizing theorems from [KS2,IK1,IK2] for arbitrary n. Corollary 1.3 With the assumptions of Theorem 1.2 in the case n ě 3 the condition dim holpM, xq ą n 2`2 n`2 for x P M implies that M is spherical in a neighborhood of the point x, and this estimate is sharp.…”

Section: Resultsmentioning

confidence: 87%

“…Then we realize in two non-equivalent ways all maximal parabolic subalgebras proving Theorem 1.4. In particular, for n " 2 we obtain an example of a CRhypersurface with dim spMq " n 2`2 n`3 " 11 that is more elementary compared to those discussed in [IK2]. For n " 4 we get an example of a CR-hypersurface with dim spMq " n 2`2 n`2 " 26; its algebra dim spMq " p 3 Ă sup3, 3q yields yet another model with symmetry of the same dimension as p 1,5 .…”

Section: Models With Large Symmetrymentioning

confidence: 86%

“…Proof of Theorem 1.2. For n " 1 the theorem was obtained in [KS2,IK1], for n " 2 its stronger variant was proven in [IK2], so we assume that n ě 3.…”

Section: Establishing the Submaximal Symmetry Dimensionmentioning

confidence: 99%

“…Finally we show other means to produce models with large symmetry: iterated blow-ups and ramified coverings. In fact, both the series of examples in Section 4.1 and those from [IK2] can be seen as a combination of a blow-up and a ramified covering.…”

Section: Models With Large Symmetrymentioning

confidence: 99%

“…Further, in the recent paper [IK2] we considered the case n " 2. It was shown that in this situation either dim spMq " 15 and M is spherical, or dim spMq ď 11 with the equality occurring only if on a dense open subset M is spherical with Levi form of signature p1, 1q.…”

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Blow-ups and infinitesimal automorphisms of CR-manifolds

Kruglikov

2020

Math. Z.

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For a real-analytic connected CR-hypersurface M of CR-dimension n ě 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra s " spM q: (i) either dim s " n 2`4 n`3 and M is spherical everywhere; (ii) or dim s ď n 2`2 n`2`δ 2,n and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version of this result is proved for the Lie group of global automorphisms of M .Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry. Introduction Formulation of the problemA classical problem in geometry is the study of symmetry algebras for classes of manifolds endowed with particular geometric structures. Given such a class C of manifolds, the symmetry algebra of M P C is the Lie algebra spMq of vector fields on M whose local flows preserve the structure, and dim spMq is called the symmetry dimension of M. An important question in this area is to determine the maximal value D max of the symmetry dimension over all M P C as well as dimensions close to it.While describing large symmetry dimensions, one often encounters a gap phenomenon, that is, the nonrealizability of some of the values immediately below D max as dim spMq for any M P C. One then searches for the next realizable value, the submaximal dimension D smax , thus obtaining the interval pD smax , D max q called the first gap, or lacuna, for the symmetry dimension. The lacunary behavior of dim spMq may extend further and, ideally, one would like to determine all possible large values of dim spMq.Mathematics Subject Classification: 32V40, 32C05; 32M12, 53C15.The best-known case for which this program has been implemented with much success, both in the global and infinitesimal settings, is Riemannian geometry [KN, Ko], see also [Eg,I2]. A large number of other situations where the gap phenomenon has been extensively studied falls in the framework of parabolic geometry [CS], see the results and historical discussion in [KT].This article concerns CR-geometry, in which case substantially less is known about the behavior of the symmetry dimension. While there was a considerable progress for Levi-nondegenerate CR-manifolds, the problem of bounding symmetry dimension in general has been wide open. The purpose of this paper is to advance in this direction.

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Section: Resultsmentioning

confidence: 87%

Section: Models With Large Symmetrymentioning

confidence: 86%

“…Proof of Theorem 1.2. For n " 1 the theorem was obtained in [KS2,IK1], for n " 2 its stronger variant was proven in [IK2], so we assume that n ě 3.…”

Section: Establishing the Submaximal Symmetry Dimensionmentioning

confidence: 99%

Section: Models With Large Symmetrymentioning

confidence: 99%

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confidence: 99%

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