arXiv research

We classify all compact simply connected homogeneous CR manifolds $M$ of codimension one and with non-degenerate Levi form up to CR equivalence. The classification is based on our previous results and on a description of the maximal connected compact group $G(M)$ of automorphisms of $M$. We characterize also the standa…

The paper constructs CR manifolds with arbitrary Levi nondegeneracy.

Study on higher order Levi forms on homogeneous CR manifolds, improving previous results.

We show that various notions of local homogeneity for CR-manifolds are equivalent. In particular, if germs at any two points of a CR-manifold are CR-equivalent, there exists a transitive local Lie group action by CR-automorphisms near every point.

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 disting…

We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.

Characterizes CR manifolds as critical points of an energy functional.

Study CR manifolds focusing on Levi and contact-nondegeneracy.

We consider a class of compact homogeneous CR manifolds, that we call $\mathfrak n$-reductive, which includes the orbits of minimal dimension of a compact Lie group $K_0$ in an algebraic homogeneous variety of its complexification $K$. For these manifolds we define canonical equivariant fibrations onto complex flag man…

An almost para-CR structure on a manifold $M$ is given by a distribution $HM \subset TM$ together with a field $K \in Γ({\rm End}(HM))$ of involutive endomorphisms of $HM$. If $K$ satisfies an integrability condition, then $(HM,K)$ is called a para-CR structure. The notion of maximally homogeneous para-CR structure of …

We extend the notion of a fundamental negatively $\mathbb Z$-graded Lie algebra $\mathfrak{m}_x=\bigoplus_{p\leq -1}\mathfrak{m}_x^p$ associated to any point of a Levi nondegenerate CR manifold to the class of $k$-nondegenerate CR manifolds $(M,\mathcal D,\mathcal J)$ for all $k\geq 2$ and call this invariant the core …

Study on CR structures in 7D, proving maximal symmetry dimension.

An explicit classification of simply connected compact homogeneous CR manifolds G/L of codimension one, with non-degenerate Levi form, is given. There are three classes of such manifolds: a) the standard CR homogeneous manifolds which are homogeneous S^1-bundles over a flag manifold F, with CR structure induced by an i…

Classifies homogeneous CR hypersurfaces in low dimensions with maximal symmetry.

Study finds maximal symmetry groups for CR structures with specific properties.

Characterizes CR manifolds in complex flag manifolds.

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.

A contact manifold $M$ can be defined as a quotient of a symplectic manifold $X$ by a proper, free action of $\R^{>0}$, with the symplectic form homogeneous of degree 2. If $X$ is, in addition, Kaehler, and its metric is also homogeneous of degree 2, $M$ is called Sasakian. A Sasakian manifold is realized naturally as …

In this paper we define the torsion flow, a CR analogue of the Ricci flow. For homogeneous CR manifolds we give explicit solutions to the torsion flow illustrating various kinds of behavior. We also derive monotonicity formulas for CR entropy functionals. As an application, we classify torsion breathers.

Develops new approach to recover CR structures from their Levi foliations.

It is shown that two Levi-Tanaka and infinitesimal CR automorphism algebras, associated with a totally nondegenerate model of CR dimension one are isomorphic. As a result, the model surfaces are maximally homogeneous and standard. This gives an affirmative answer in CR dimension one to a certain question formulated by …

The paper classifies CR structures on 3D Lie groups, focusing on SL2(R).

We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kaehler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for w…

Classifies CR submanifolds in complex hyperbolic spaces.

The paper explores CR structures and their leaf spaces in semi-Riemannian manifolds.

Study CR submanifolds of nearly Kähler S³ × S³ using properties of its almost product structure.

E. Cartan's method of moving frames is applied to 3-dimensional manifolds $M$ which are CR-embedded in 5-dimensional real hyperquadrics $Q$ in order to classify $M$ up to CR symmetries of $Q$ given by the action of one of the Lie groups $SU(3,1)$ or $SU(2,2)$. In the latter case, the CR structure of $M$ derives from a …

Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the `quaternionic-like manifolds'. These contain, as particular subclasses, the CR quaternionic and the $ρ$-quaternionic manifolds. Moreover, …

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for $CR$ manifolds and Hörmander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial diffe…

A CR manifold $M$, with CR distribution $\mathcal D^{10}\subset T^\mathbb C M$, is called {\it totally nondegenerate of depth $μ$} if: (a) the complex tangent space $T^\mathbb C M$ is generated by all complex vector fields that might be determined by iterated Lie brackets between at most $μ$ fields in $\mathcal D^{10} …

We consider the significant class of holomorphically nondegenerate CR manifolds of finite type that are represented by some weighted homogeneous polynomials and we derive some useful features which enable us to set up a fast effective algorithm to compute their Lie algebras of infinitesimal CR-automorphisms. This algor…

Extends potential theory to Carnot groups, estimating Hausdorff dimension.

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions $n$ and codimensions $n^2$ are among the very few possibilities of the so called parabolic geometries. Indeed, the homogene…

We study non-degenerate CR geometries of hypersurface type that are symmetric in the sense that, at each point, there is a CR transformation reversing the CR distribution at that point. We show that such geometries are either flat or homogeneous. We show that non-flat non-degenerate symmetric CR geometries of hypersurf…

Defines pre-Kähler structures and their properties.

In this article, we solve the equivalence problem for 2--nondegenerate CR geometries that have (at every point) a homogeneous space $G/H$ as a maximally symmetric model for $G$ simple real Lie group of CR automorphisms. This completes the classification of real submanifolds in complex space that are maximally symmetric…

Study para-CR structures relaxing complex conjugation constraint, finding homogeneous models.

We study CR hypersurfaces in C^4 with constant rank Levi form and find their defining equations.

We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced. It measures how a deformation deviates from a one parameter family of motions u…

Classifies homogeneous Riemannian structures on 3D Lie groups.

Study CR Yamabe constant and CR structures on manifolds.

Study CR Yamabe constant, flow, and soliton on CR manifolds.

New CR hypersurfaces in complex space with specific properties.

Study of CR-submanifolds in various Lorentzian manifolds.

Study on CR Paneitz operator on non-embeddable CR manifolds.

In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the composition series and branching problems for objects in the Bernstein-Gelfand-Gelfand pa…

Survey on CR Paneitz operator and manifold embeddability.

The nonnegativity of the CR Paneitz operator plays a crucial role in three-dimensional CR geometry. In this paper, we prove this nonnegativity for embeddable CR manifolds. This result and previous works give an affirmative solution of the CR Yamabe problem for embeddable CR manifolds. We also show the existence of a co…