Blow-ups and infinitesimal automorphisms of CR-manifolds

Abstract

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.

New examples of 2‐nondegenerate real hypersurfaces in CN$\mathbb {C}^N$ with arbitrary nilpotent symbols

“…is in the right kernel of 𝐻 (i.e., 𝐻𝑣 = 0). A matrix representing one of the Levi kernel adjoint operators of 𝑀 (respectively, 𝑀 𝑗 ) is given by ( 8) with 𝑣 as in (17) (respectively, (15)), from which it is clearly seen that the matrix is a direct sum of matrices representing nonzero Levi kernel adjoint operators of 𝑀 1 and 𝑀 2 . Hence, 𝑀 = {ℜ(𝑧 0 ) = Φ} is uniformly 2-nondegenerate with a rank 1 Levi kernel.…”

“…We expect that many more such examples exist, although deriving their closed form descriptions will be a challenge as they arise from solutions to rather complicated overdetermined system of partial differential equations (PDEs). For 𝑘 > 2, there are examples of homogeneous structures (e.g., [6,8,9,16,18,20]) and the recent study [17] of nonhomogeneous 3-nondegenerate structures in ℂ 4 . Given the early state of this 𝑘 > 2 frontier, the analogous open problems of finding closed form defining equations are interesting for higher orders of 𝑘-nondegeneracy.…”

On 3-nondegenerate CR manifolds in dimension 7 (I): the transitive case

“…Further results in this direction for n = 1 (sphericity of M near x if dim inf(M, D, J ; x) > 5) and n = 2 (sphericity of M near x if dim inf(M, D, J ; x) > 11) are contained in [13] and [11] respectively. (For related results in the case n > 2 see [14].) Our goal is to prove Beloshapka's conjecture for n = 3.…”

“…Due to this reason, we split the presentation of our results in two separate works, cf. [15]. This part (I) is dedicated to the locally transitive case.…”