Homogeneous CR Manifolds Research Articles

A characterization of homogeneous three-dimensional CR manifolds

We characterize homogeneous three-dimensional CR manifolds, in particular Rossi spheres, as critical points of a certain energy functional that depends on the Webster curvature and torsion of the pseudohermitian structure.

Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on CR manifolds

We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented pseudohermitian structures \mathcal{P}_+ . We show that the functionals are continuous with respect to a natural topology on \mathcal{P}_+ . Using an adaptation of the standard Kato–Rellich perturbation theory, we prove that the functionals are (one-sided) differentiable along 1-parameter analytic deformations. We use this differentiability to define the notion of critical pseudohermitian structures, in a generalized sense, for them. We give a necessary (also sufficient in some situations) condition for a pseudohermitian structure to be critical. Finally, we present explicit examples of critical pseudohermitian structures on both homogeneous and non-homogeneous CR manifolds.

On finitely nondegenerate closed homogeneous CR manifolds

A complex flag manifold $$\textsf {F}{=}{{\textbf {G}}}/{{\textbf {Q}}}$$ decomposes into finitely many real orbits under the action of a real form $${{\textbf {G}}}^\upsigma $$ of $${{\textbf {G}}}$$ . Their embedding into $$\textsf {F}$$ defines on them CR manifold structures. We characterize and list all the closed real orbits which are finitely nondegenerate.

Compact homogeneous Leviflat CR-manifolds

We consider compact Leviflat homogeneous Cauchy–Riemann (CR) manifolds. In this setting, the Levi-foliation exists and we show that all its leaves are homogeneous and biholomorphic. We analyze separately the structure of orbits in complex projective spaces and parallelizable homogeneous CR-manifolds in our context and then combine the projective and parallelizable cases. In codimensions one and two, we also give a classification.

Higher order Levi forms on homogeneous CR manifolds

We investigate the nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous CR manifolds. Improving previous results, we prove that general orbits of real forms in complex flag manifolds have order less or equal than 3 and the compact ones less or equal 2. Finally we construct by Lie extensions weakly nondegenerate CR vector bundles with arbitrary orders of nondegeneracy.

Holomorphically homogeneous CR-manifolds and their model surfaces

Показано, что модельная поверхность ростка голоморфно однородного CR-многообразия также голоморфно однородна. Для типа по Блуму-Грэму ростка голоморфно однородного CR-многообразия получены ограничения на кратности. Библиография: 6 наименований.

Holomorphically homogeneous CR-manifolds and their model surfaces

We show that the model surface of a germ of a holomorphically homogeneous CR-manifold is holomorphically homogeneous. We also obtain restrictions on the multiplicities in the Bloom–Graham type of a germ of a holomorphically homogeneous CR-manifold.

On transitive contact and CR algebras

We consider locally homogeneous CR manifolds and show that, under a condition only depending on their underlying contact structure, their CR automorphisms form a finite dimensional Lie group

Mostow’s fibration for canonical embeddings of compact homogeneous $CR$ manifolds

We define a class of compact homogeneous CR manifolds which are bases of Mostow fibrations having total spaces equal to their canonical complex realizations and Hermitian fibers. This is used to establish isomorphisms between their tangential Cauchy–Riemann cohomology groups and the corresponding Dolbeault cohomology groups of the embeddings.

Mostow’s fibration for canonical embeddings of compact homogeneous CR manifolds

We define a class of compact homogeneous CR manifolds which are bases of Mostow fibrations having total spaces equal to their canonical complex realizations and Hermitian fibers. This is used to establish isomorphisms between their tangential Cauchy–Riemann cohomology groups and the corresponding Dolbeault cohomology groups of the embeddings.

Lie algebras of infinitesimal CR automorphisms of weighted homogeneous and homogeneous CR-generic submanifolds of CN

We consider the significant class of homogeneous CR manifolds represented by some weighted homogeneous polynomials and we derive some plain and useful features which enable us to set up a fast effective algorithm to compute homogeneous components of their Lie algebras of infinitesimal CR automorphisms. This algorithm mainly relies upon a natural gradation of the sought Lie algebras, and it also consists in treating separately the related graded components. While classical methods are based on constructing and solving some associated PDE systems which become time consuming as soon as the number of variables increases, the new method presented here is based on plain techniques of linear algebra. Furthermore, it benefits from a divide-and-conquer strategy to break down the computations into some simpler subcomputations. Also, we consider the new and effective concept of comprehensive Gr?bner systems which provides us some powerful tools to treat the computations in the much complicated parametric case. The designed algorithm is also implemented in the Maple software, what required also implementing a recently designed algorithm of Kapur et al.

REDUCTIVE COMPACT HOMOGENEOUS CR MANIFOLDS

We define and investigate a class of compact homogeneous CR manifolds, that we call $$ \mathfrak{n} $$ -reductive. They are orbits of minimal dimension of a compact Lie group K 0 in algebraic affine homogeneous spaces of its complexification K. For these manifolds we obtain canonical equivariant fibrations onto complex flag manifolds, generalizing the Hopf fibration $$ {S^3}\to \mathbb{C}{{\mathbb{P}}^1} $$ . These fibrations are not, in general, CR submersions, but satisfy the weaker condition of being CR-deployments; to obtain CR submersions we need to strengthen their CR structure by lifting the complex stucture of the base.

Classification of homogeneous CR-manifolds in dimension 4

Locally homogeneous CR-manifolds in dimension 3 were classified, up to local CR-equivalence, by E. Cartan. We classify, up to local CR-equivalence, all locally homogeneous CR-manifolds in dimension 4. The classification theorem enables us also to classify all symmetric CR-manifolds in dimension 4, up to local CR-equivalence.

Fibrations and globalizations of compact homogeneous CR-manifolds

Fibration methods which were previously used for complex homogeneous spaces and CR-homogeneous spaces of special types [1]-[4] are developed in a general framework. These include the -anticanonical fibration in the CR-setting, which reduces certain considerations to the compact projective algebraic case, where a Borel-Remmert type splitting theorem is proved. This leads to a reduction to spaces homogeneous under actions of compact Lie groups. General globalization theorems are proved which enable one to regard a homogeneous CR-manifold as an orbit of a real Lie group in a complex homogeneous space of a complex Lie group. In the special case of CR-codimension at most two, precise classification results are proved and are applied to show that in most cases there exists such a globalization.

Complex vector fields and hypoelliptic partial differential operators

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for CR manifolds, that was first introduced by two of the authors; and the Hormander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of (0,1) vector fields satisfies a subelliptic estimate.

Classification of Levi degenerate homogeneous CR-manifolds in dimension 5

In this paper we present new examples of homogeneous 2-nondegenerate CR-manifolds of dimension 5 and give, up to local CR-equivalence, a full classification of all CR-manifolds of this type.

Locally homogeneous finitely nondegenerate CR-manifolds

Germs of locally homogeneous CR manifolds M can be characterized in terms of certain algebraic data, e.g., by CR-algebras. We give an explicit formula which relates the Levi form of such an M and its higher order analogues to the Lie brackets in certain finite dimensional Lie algebras. As an application we give a simple characterization of geometric properties of M such as minimality, k-nondegeneracy, holomorphic degeneracy etc. in purely algebraic terms. We present an example of a homogeneous 3-nondegenerate CR-manifold. We also determine a universal upper bound for the order k of k-nondegenerate orbits of real forms in flag manifolds.

ON HOMOGENEOUS CR MANIFOLDS AND THEIR CR ALGEBRAS

In this paper we show some results on homogeneous CR manifolds, proved by introducing their associated CR algebras. In particular, we give different notions of nondegeneracy (generalizing the usual notion for the Levi form) which correspond to geometrical properties for the corresponding manifolds. We also give distinguished equivariant CR fibrations for homogeneous CR manifolds. In the second part of the paper we apply these results to minimal orbits for the action of a real form of a semisimple Lie group Ĝ on a flag manifold Ĝ/Q.

Algebras of infinitesimal CR automorphisms

This paper is devoted to the investigation of Lie algebras of local infinitesimal CR automorphisms. Such algebras are naturally associated to germs of homogeneous CR manifolds. The authors introduce a corresponding abstract notion of CR algebra. A CR algebra is a pair $(L,L_1)$(L,L1), consisting of a real Lie algebra $L$L and a subalgebra $L_1$L1 of the complexification $\bold C\otimes_{\bold R} L$C⊗RL, such that the factor space $L/L\cap L_1$L/L∩L1 is finite-dimensional. The authors investigate some formal properties of CR algebras and construct some fibrations'' (i.e., $L$L-equivariant submersions) of such algebras. They introduce three new notions of nondegeneracy of CR algebras---strict, weak and ideal nondegeneracy. These three concepts are weaker than those used previously by some other authors. The authors intend to extend the application of the E. Cartan method of investigating the equivalence of CR structures to some larger classes of CR manifolds. One of the main ideas of this paper is a decomposition of arbitrary CR algebras into three parts'': totally real, totally complex and weakly nondegenerate CR algebras (Theorems 5.3 and 5.4). There are some results about these three special classes of CR algebras. Some results about prolongations for transitive CR algebras are also obtained, in particular about maximality of parabolic CR algebras with respect to transitive prolongations.

On the local equivalence of homogeneous CR-manifolds

For certain classes of homogeneous Cauchy-Riemann manifolds it is shown that the local CR-equivalence problem is equivalent to the global CR-equivalence problem. Among these are large classes of compact linear group orbits.