CR Manifolds Research Articles

Equivariant Kodaira Embedding for CR Manifolds with Circle Action

We consider a compact CR manifold with a transversal CR locally free circle action endowed with an S1-equivariant positive CR line bundle. We prove that a certain weighted Fourier–Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion. As a consequence, we establish an equivariant Kodaira embedding theorem.

Flag structures on real 3-manifolds

We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are given by totally real submanifolds in the flag space $$\mathbf{SL}(3,{{\mathbb {C}}})/B$$ , where B is the subgroup of upper triangular matrices. We also define a global invariant which is analogous to the Chern–Simons secondary class invariant for three manifolds with a Riemannian structure and to the Burns–Epstein invariant in the case of CR structures. It turns out to be constant on homotopy classes of totally real immersions in flag space.

An integral type Brezis-Nirenberg problem on the Heisenberg group

This paper is devoted to a class of integral type Brezis-Nirenbreg problems on the Heisenberg group. They are a class of nonlinear integral equations on the bounded domains of Heisenberg group and related to the CR Yamabe problems on CR manifold. Based on the sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group, the nonexistence and existence results are obtained.

Locally CR spherical three manifolds

Every open and orientable three manifold has a CR structure which is locally equivalent to the standard CR structure on S 3 S^3 .

Aerobatics of Flying Saucers

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer’s velocity. These restrictions define certain ‘manœuvres’ of the saucer, which we call ‘attacking,’ ‘landing,’ or ‘$$G_2$$ mode’ manœuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manœuvre corresponds to the flat Legendrean contact structure, the landing manœuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1, 1), and the most complicated $$G_2$$ manœuvre corresponds to the contact Engel structure (Engel in C R Acad Sci 116:786–788, 1893; Mano et al. in The geometry of marked contact twisted cubic structures, 2018, arXiv:1809.06455) with split real form of the exceptional Lie group $$G_2$$ as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic $$G_2$$ geometries is used to construct a ‘$$G_2$$ joystick,’ consisting of two balls of radii in ratio $$1\!:\!3$$ that transforms the difficult $$G_2$$ manœuvre into the pilot’s action of rolling one of joystick’s balls on the other without slipping nor twisting.

Aerodynamics of Flying Saucers

We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M. Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With yet another structure on M, the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of ‘parabolic geometry’) and we provide examples when this geometry is ‘flat,’ meaning that its symmetries comprise the split form of the exceptional Lie algebra $${\mathfrak {g}}_2$$.

Morse inequalities for Fourier components of Kohn–Rossi cohomology of CR covering manifolds with S1-action

Let $X$ be a compact connected CR manifold of dimension $2n+1, n \geq 1$. Let $\widetilde{X}$ be a paracompact CR manifold with a transversal CR $S^1$-action, such that there is a discrete group $\Gamma$ acting freely on $\widetilde{X}$ having $X \, = \, \widetilde{X}/\Gamma$. Based on an asymptotic formula for the Fourier components of the heat kernel with respect to the $S^1$-action, we establish the Morse inequalities for Fourier components of reduced $L^2$-Kohn-Rossi cohomology with values in a rigid CR vector bundle over $\widetilde{X}$. As a corollary, we obtain the Morse inequalities for Fourier components of Kohn-Rossi cohomology on $X$ which were obtained by Hsiao-Li by using Szeg\"{o} kernel method.

Effect of oxygen deficiency on optical and magnetic properties of Ba2MMoO6 (M=Cr, Mn, Fe): A first-principles study

Molybdenum based Ba2MMoO6 double perovskite structures have been known to show half-metallic properties which can lead to spintronic applications. By introducing point defects in oxygen sites, the variation of optical and magnetic properties of cubic Ba2MMoO6 (M = Cr, Mn, and Fe) structures have been calculated within the DFT framework in this study by using the GGA-PBE functional. The structures were found to give a ferromagnetic ground state. There is a reduction in magnetic strength for both Ba2MnMoO6 and Ba2FeMoO6 structures when defects are introduced but it increases for the Ba2CrMoO6 structure. As we add defects to the structures, the half-metallicity is slightly reduced which affects any potentiality and suitability in spintronic applications. It has been found that the dielectric function is extremely high for the pure Ba2FeMoO6 which leads to a very high refractive index for zero incident photon energy. Furthermore, most of the photon absorption for these structures occurs in the ultraviolet region but the addition of oxygen defects may enhance the optical properties of Ba2MnMoO6 double perovskite compound for visible photons.

Semi-isometric CR immersions of CR manifolds into Kähler manifolds and applications

We study the second fundamental form of semi-isometric CR immersions from strictly pseudoconvex CR manifolds into K\"ahler manifolds. As an application, we give a precise condition for the CR umbilicality of real hypersurfaces, extending an well-known theorem by Webster on the nonexistence of CR umbilical points on generic real ellipsoids. As other applications, we extend the linearity theorem of Ji-Yuan for CR immersions into spheres with vanishing second fundamental form to the important case of three-dimensional manifolds, and prove the ``first gap'' theorem in the spirit of Webster, Faran, Cima-Suffridge, and Huang for semi-isometric CR immersions into a complex euclidean space of ``low'' codimension. Our new approach to the linearity theorem is based on the study of the first positive eigenvalue of the Kohn Laplacian.

A Constraint on Chern Classes of Strictly Pseudoconvex CR Manifolds

This short paper gives a constraint on Chern classes of closed strictly pseudoconvex CR manifolds (or equivalently, closed holomorphically fillable contact manifolds) of dimension at least five. We also see that our result is ''optimal'' through some examples.

A metal lift-off process through hyperbolic undercut of laser heat-mode lithography

Laser heat-mode lithography is an important method for the fabrication of micro and nanostructure. A metal lift-off method through hyperbolic undercut of laser heat-mode lithography was proposed. In this method, a desirable hyperbolic undercut profile was used to replace the inverted trapezoid undercut, and the hyperbolic undercut profile can be obtained through laser heat-mode lithography technique. Cr structures of gratings, grids and planar spiral inductor with a minimum linewidth of 0.42 μm and a height of 100 nm were obtained successfully. These indicate that the lift-off process through hyperbolic undercut of laser heat-mode lithography can be used to fabricate the submicron metal structures.

Slant Curves in Contact Lorentzian Manifolds with CR Structures

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the ∇ ^ -geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when c ≤ 0 , there does not exist a non-geodesic slant Frenet curve satisfying the ∇ ^ -Jacobi equations for the ∇ ^ -geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms M 1 3 ( H ^ ) for H ^ = 2 c > 0 .

Sufficient and necessary conditions for local rigidity of CR mappings and higher order infinitesimal deformations

In this paper we continue our study of local rigidity for maps of CR submanifolds of the complex space. We provide a linear sufficient condition for local rigidity of finitely nondegenerate maps between minimal CR manifolds. Furthermore, we show higher order infinitesimal conditions can be used to give a characterization of local rigidity.

Stability of the existence of a pseudo-Einstein contact form

A pseudo-Einstein contact form plays a crucial role in defining some global invariants of closed strictly pseudoconvex CR manifolds. In this paper, we prove that the existence of a pseudo-Einstein contact form is preserved under deformations as a real hypersurface in a fixed complex manifold of complex dimension at least three.

The □B-Heat Equation on CR Manifolds of Finite Type with Comparable Levi Forms

We study the initial-value problems for the heat equations associated with the operator □b on compact CR manifolds of finite type. The critical component of our analysis is the condition called Dϵ(q) and introduced by K. D. Koenig [Amer. J. Math., 124, 129–197 (2002)]. Actually, it states that the min{q, n − 1 − q}th smallest eigenvalue of the Levi form is comparable with the largest eigenvalue of the Levi form.

Regularity of CR mappings of abstract CR structures

We study the [Formula: see text] regularity problem for CR maps from an abstract CR manifold [Formula: see text] into some complex Euclidean space [Formula: see text]. We show that if [Formula: see text] satisfies a certain condition called the microlocal extension property, then any [Formula: see text]-smooth CR map [Formula: see text], for some integer [Formula: see text], which is nowhere [Formula: see text]-smooth on some open subset [Formula: see text] of [Formula: see text], has the following property: for a generic point [Formula: see text] of [Formula: see text], there must exist a formal complex subvariety through [Formula: see text], tangent to [Formula: see text] to infinite order, and depending in a [Formula: see text] and CR manner on [Formula: see text]. As a consequence, we obtain several [Formula: see text] regularity results generalizing earlier ones by Berhanu–Xiao and the authors (in the embedded case).

On the Pseudohermitian Curvature of Contact Semi-Riemannian Manifolds

Let M be a contact semi-Riemannian manifold, equivalently a non degenerate almost CR manifold. In this paper we study the pseudo-hermitian Ricci curvature, pseudo-Einstein and $$\eta $$-Einstein manifolds. Then, by using the pseudo-Einstein and the $$\eta $$-Einstein conditions, some rigidity theorems are established to characterize Sasakian manifolds among nondegenerate CR manifolds. In particular, if the Webster metric $$g_\theta $$ of nondegenerate CR structure $$({\mathcal {H}},\theta ,J)$$ is pseudo-Einstein with Webster scalar curvature $${\hat{r}}\ne 0$$, then there exists a real constant $$t\ne 0$$ for which the Webster metric associated to $$({\mathcal {H}},t\theta ,J)$$ is Einstein–Sasakian.

The Asymptotics of the $L^2$-analytic Torsion on CR Manifolds with $S^1$ Action

The Asymptotics of the $L^2$-analytic Torsion on CR Manifolds with $S^1$ Action

On the geometric order of totally nondegenerate CR manifolds

A CR manifold M, with CR distribution $$\mathcal {D}^{10} \subset T^\mathbb {C}M$$ , is called totally nondegenerate of depth $$\mu $$ 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 $$\mu $$ fields in $$\mathcal {D}^{10} + \overline{\mathcal {D}^{10}}$$ ; (b) for each integer $$2 \le k \le \mu -1$$ , the families of all vector fields that might be determined by iterated Lie brackets between at most k fields in $$\mathcal {D}^{10} + \overline{\mathcal {D}^{10}}$$ generate regular complex distributions; (c) the ranks of the distributions in (b) have the maximal values that can be obtained amongst all CR manifolds of the same CR dimension and satisfying (a) and (b)—this maximality property is the total nondegeneracy condition. In this paper, we prove that, for any Tanaka symbol $$\mathfrak {m}= \mathfrak {m}^{-\mu } + \cdots + \mathfrak {m}^{-1}$$ of a totally nondegenerate CR manifold of depth $$\mu \ge 4$$ , the full Tanaka prolongation of $$\mathfrak {m}$$ has trivial subspaces of degree $$k \ge 1$$ , i.e. it has the form $$\mathfrak {m}^{-\mu } + \cdots + \mathfrak {m}^{-1} + \mathfrak {g}^0$$ . This result has various consequences. For instance it implies that any (local) CR automorphism of a regular totally nondegenerate CR manifold is uniquely determined by its first order jet at a fixed point of the manifold. It also gives a complete proof of a conjecture by Beloshapka on the group of automorphisms of homogeneous totally nondegenerate CR manifolds.

Curvature Inequalities for Submanifolds of S-space From

In this paper we establish new results of squared mean curvature and Ricci curvature for the sub manifolds of S-space from that is the generalization of complex and contact structures. Obtained results are discussed for invariant, anti invariant and CR sub manifolds of S-space from.