arXiv research

Study CR Yamabe constant and CR structures on manifolds.

CR structure on S³ with non-compact solutions to CR Yamabe problem.

Study on CR structures on 3D Lie groups, focusing on equivalence and closed chains.

Study infinitesimal CR symmetries of accidental CR structures.

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 chains in 3D CR and para-CR structures.

Class I CR manifolds have initial G-structure a certain 4-dimensional subgroup of GL_3(C). Class II CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_4(C). Class III-1 CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_5(C). Class III-2 CR manifolds have initial G-…

The study proves CR structures on specific three-manifolds are equivalent to standard structures.

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

Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.

Defines Lewy curves in para-CR geometry and characterizes their path geometries.

Study of CR-submanifolds in various Lorentzian manifolds.

Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster con…

We develop a spinorial description of CR structures of arbitrary codimension. More precisely, we characterize almost CR structures of arbitrary codimension on (Riemannian) manifolds by the existence of a Spin$^{c, r}$ structure carrying a partially pure spinor field. We study various integrability conditions of the alm…

Compactifies CR structures for complex hyperbolic manifolds.

We classify the normal CR structures on $S^3$ and their automorphism groups. Together with [3], this closes the classification of normal CR structures on contact 3-manifolds. We give a criterion to compare 2 normal CR structures, and we show that the underlying contact structure is, up to homotopy, unique.

CR Killing operator derived from tractor calculus for CR structures.

The paper studies stability of CR structures on compact manifolds.

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

Let M be a G2-manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy G2 manifold. We interpret G2-instanton bundles as CR-holomorphic bundles on its twistor space.

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 propose two constructions extending the Chern-Moser normal form to non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One of them translates the Chern-Moser normalization into pure intrinsic setting, whereas the other directly extends the (extrinsic) Chern-Moser normal form by allowing non-C…

Study reveals CR structure of snake robot's geometry.

Study on surface geometry in Lie groups with CR structures.

This paper demostrates a method for analysing almost CR geometries $(H,J)$, by uniquley defining a partially integrable structure $(H,K)$ from the same data. Thus two almost CR geometries $(H,J)$ and $(H',J')$ are equivalent if and and only if they generate isomorphic induced partially integrable CR geometries $(H,K)$ …

In this paper, we solve the so-called CR Poincaré-Lelong equation by solving the CR Poisson equation on a complete noncompact CR $(2n+1)$-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of Kähler geometry. With applications of this sol…

The purpose of this paper is to introduce a geometric structure called pseudo-conformal quaternionic CR structure on a (4n+3)-dimensional mamnifold and then exhibit a quaternionic analogue of Chern-Moser's CR structure and uniformization.

We construct a generalization of Courant algebroids which are classified by the third cohomology group $H^3(A,V)$, where $A$ is a Lie Algebroid, and $V$ is an $A$-module. We see that both Courant algebroids and $\mathcal{E}^1(M)$ structures are examples of them. Finally we introduce generalized CR structures on a manif…

Suppose $M_{1}$ and $M_{2}$ are $3$-dimensional closed (compact without boundary) CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of $M_{1}$ and $M_{2}$ also admits a CR structure with positive CR Yamabe constant.

We describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R^2 x R^2. We introduce the notion of a finite type singularity analogous to CR geometry and, al…

Suppose $M_{1}$ and $M_{2}$ are two closed (compact with no boundary) spherical CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of $M_{1}$ and $M_{2}$ also admits a spherical CR structure with positive CR Yamabe constant.

Unified CR-twistor spaces for $G_2$ and $Spin(7)$ structures.

Let M^3 be a closed CR 3-manifold. In this paper we derive a Bochner formula for the Kohn Laplacian in which the pseudo-hermitian torsion plays no role. By means of this formula we show that the non-zero eigenvalues of the Kohn Laplacian are bounded below by a positive constant provided the CR Paneitz operator is non-n…

New CR manifolds found with same Kohn Laplacian spectra.

Let $Ω$ be a bounded strictly pseudoconvex domain in $C^2$ with a smooth, connected and compact boundary M and having a CR structure $J_0$ induced from $C^2$. Assume this CR structure has zero Webster torsion. Then if we deform the CR structure through real-analytic dependence on the deformation parameter and such that…

We establish Bochner-type formulas for operators related to $CR$ automorphisms and spherical $CR$ structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.

The paper calculates variations of Einstein-Hilbert action on CR manifolds.

The systematic study of CR manifolds originated in two pioneering 1932 papers of Élie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR structures on $\mathrm{SU}_2 = S^3$, deforming the standard `spherical' structure…

In this paper we construct examples of $CR$ deformations of Lorentzian hypersurfaces which are $CR$ embeddable at all points outside an arbitrarily small compact set whose interior contains a point where $CR$ embeddablity is not possible.

We define a CR structure on a distinguished hyperplane in $\mathbb{C}^{n+1}$ and the CR sub-Laplacian on this CR manifold. We also define symmetries of the CR sub-Laplacian in general and for this special case construct all of them using the ambient construction. Then we investigate the algebra structure of the symmetr…

New approach linking CR Yamabe invariant to Sasaki structures.

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

3D projective structures can be metrized with conformal structures.

We explicitly determine the structure equations of 5-dimensional Levi 2-nondegenerate CR hypersurfaces, using our recently constructed canonical Cartan connection for this class of CR manifolds. We also give an outline of the basic properties of absolute parallelisms and Cartan connections, together with a brief discus…

We introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a 3-dimensional CR-structure, which we call quasi-Fefferman metrics. These metrics generalise the Fefferman metric but allow for more control of the Ricci curvature. Our main result is a criterion for embaddability of 3-dimensional CR-struc…

We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotiens of the 3-sphere or of a circle bundle over a Riemann surface of positive genus.…

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…

We consider a class of stratified groups with a CR structure and a compatible control distance. For these Lie groups we show that the space of conformal maps coincide with the space of CR and anti-CR diffeomorphisms. Furthermore, we prove that on products of such groups, all CR and anti-CR maps are product maps, up to …