CR Manifolds Research Articles

A Frankel type theorem for CR submanifolds of Sasakian manifolds

A Frankel type theorem for CR submanifolds of Sasakian manifolds

Webster pseudo-torsion formulas of CR manifolds

Webster pseudo-torsion formulas of CR manifolds

A heat equation approach to intertwining

In this paper we present anew approach based on the heat equation and extension problems to some intertwining formulas arising in conformal CR geometry.

On holomorphic foliations admitting invariant CR manifolds

We study holomorphic foliations of codimension $k\geq 1$ on a complex manifold $X$ of dimension $n+k$ from the point of view of the exceptional minimal set conjecture. For $n\geq 2$ we show in particular that if the holomorphic normal bundle $N_{\mathcal{F}}$ is Griffiths positive, then the foliation does not admit a compact invariant set that is a complete intersection of $k$ smooth real hypersurfaces in $X$.

Toeplitz Operators on CR Manifolds and Group Actions

Let X be a connected orientable compact CR manifold with non-degenerate Levi form. In this paper, we study the algebra of Toeplitz operators on X and we establish star product for a certain class of symbols on X. In the second part of this paper, we consider a locally free action of a compact Lie group G on X and we investigate the associated algebra of G-invariant Toeplitz operators. The main application is a construction of deformation quantization of compact “quantizable” pseudo-Kähler manifolds.

Geometry of Warped Product CR and Semi-Slant Submanifolds in Quasi-Para-Sasakian Manifolds

In the present paper we study the existence or non-existence of warped product semi-slant submanifolds in quasi-para-Sasakian manifolds and prove that there are no proper warped product semi-slant submanifolds in a quasi-para-Sasakian manifold such that totally geodesic and totally umbilical submanifolds of warped product are proper semi-slant and invariant (or anti-invariant), respectively.

Heat kernel asymptotics for Kohn Laplacians on CR manifolds

Let X be an abstract orientable (not necessarily compact) CR manifold of dimension 2n+1, n≥1, and let Lk be the k-th tensor power of a CR complex line bundle L over X. Suppose that condition Y(q) holds at each point of X, we establish asymptotics of the heat kernel of Kohn Laplacian with values in Lk. As an application, we give a heat kernel proof of Morse inequalities on compact CR manifolds. When X admits a transversal CR R-action, we also establish asymptotics of the R-equivariant heat kernel of Kohn Laplacian with values in Lk. As an application, we get R-equivariant Morse inequalities on compact CR manifolds with transversal CR R-action.

G-Equivariant Embedding Theorems for CR Manifolds of High Codimension

Let (X,T1,0X) be a (2n+1+d)-dimensional compact CR manifold with codimension d+1, d≥1, and let G be a d-dimensional compact Lie group with CR action on X and T be a globally defined vector field on X such that CTX=T1,0X⊕T0,1X⊕CT⊕Cg_, where g_ is the space of vector fields on X induced by the Lie algebra of G. In this work, we show that if X is strongly pseudoconvex in the direction of T and n≥2, then there exists a G-equivariant CR embedding of X into CN for some N∈N. We also establish a CR orbifold version of Boutet de Monvel’s embedding theorem.

Implicit Bias Scenario Design: What Can We Learn from Cognitive Science?

Designing simulation-based experiences (SBE) that address bias requires an understanding of evidence-based bias mitigating strategies to accurately identify critical actions and performance measures. However, mitigating bias strategies have their own set of ambiguities. Through the lens of applied behavioral and cognitive science, this paper presents an interdisciplinary framework on “perspective-taking” that can be used to explore mechanisms of implicit bias, bias mitigating strategies, and practical applications that can be incorporated into simulation-based experiences. By analyzing the interconnections between attention, judgment, and the reconstruction of experience, we specify at least two strategies for implicit bias SBE. We propose that attention-directing strategies can be incorporated into scenario design through: “Cognitive Recreation” (CR), with a focus on mirror neuron activation in recreating others’ experiences; and “Emotional Distance Reflection” (EDR), with a focus on minimizing emotional bias, for example, egocentricity bias in the emotional domain (EEB). We specify CR and EDR structure and application in the Keyser-Gilbert Framework.

Bernstein spaces on Siegel CR manifolds

In this paper we introduce and study Bernstein spaces on a class of quadratic CR manifolds, which we call Siegel CR manifolds. These are spaces of entire functions of exponential type whose restrictions to a given Siegel CR submanifold are \(L^p\)-integrable with respect to a natural measure. For these spaces, among other results, we prove the Plancherel–Pólya inequality, a Bernstein inequality and a sufficient condition for a sequence to be sampling.

Szegő Kernel Asymptotics on Complete Strictly Pseudoconvex CR Manifolds

We prove a Bochner–Kodaira–Nakano formula and establish Szegő kernel expansions on complete strictly pseudoconvex CR manifolds with transversal CR mathbb {R}-action under certain natural geometric conditions. As a consequence we show that such manifolds are locally CR embeddable.

An upper bound for the first positive eigenvalue of the Kohn Laplacian on Reinhardt real hypersurfaces

A real hypersurface in C 2 \mathbb {C}^2 is said to be Reinhardt if it is invariant under the standard T 2 \mathbb {T}^2 -action on C 2 \mathbb {C}^2 . Its CR geometry can be described in terms of the curvature function of its “generating curve”, i.e., the logarithmic image of the hypersurface in the plane R 2 \mathbb {R}^2 . We give a sharp upper bound for the first positive eigenvalue of the Kohn Laplacian associated to a natural pseudohermitian structure on a compact and strictly pseudoconvex Reinhardt real hypersurface having closed generating curve (which amounts to the T 2 \mathbb {T}^2 -action being free). Our bound is expressed in terms of the L 2 L^2 -norm of the curvature function of the generating curve and is attained if and only if the curve is a circle.

A non-Hausdorff space of CR equivalence classes

Examples show that in three dimensions the natural topology on the space of equivalence classes of CR structures need not be Hausdorff. However, the space of equivalence classes of compact aspherical CR structures is Hausdorff. Thus the non-Hausdorff property is due to spherical or Levi-flat points.

The Influence of Anion-Stripped MIL-101(Cr) Dispersed in Thin-Film Polyvinyl Alcohol Membrane Matrix on the Methylene Blue Dye Separation

This study investigates the production and performance of a novel nanofiltration membrane for removal of cationic dye (Methylene blue) and multivalent cations. These positively charged membranes are made by dispersing a modified cationic metal–organic framework, Cl-MIL-101(Cr), into the polyvinyl alcohol matrix as a membrane skin layer. To this end, the mobile anion (Cl−) embedded in the MIL-101(Cr) structure plays a role to create a positive partial charge on the membrane. In this study, the effects of MOF content and their types on the membrane structure were considered by FTIR, XRD, FESEM, Zeta potential and water contact angle. The results have shown that membranes filled with Cl-MIL-101(Cr) attained higher permeate flux and rejection than those of MIL-101(Cr). Particularly, this study indicates that the low irreversible resistance (19.49%) and high flux return ratio (80.50%) have been related to the membrane containing 15% cationic Cl-MIL-101(Cr). However, this membrane rejected more than 30.41% of AlCl3 salt and 99.08% of methylene blue with approximate permeate flux of 20 L/m2·h. It is recommended that the fabricated membrane be placed in the flow path process of cationic dyes purification.

The 1-jet determination of stationary discs attached to generic CR submanifolds

The existence of a non-defective stationary disc attached to a non-degenerate model quadric in \({\mathbb {C}}^N\) is a necessary condition to ensure the unique 1-jet determination of the lifts of a key family of stationary discs [6]. In this paper, we give an elementary proof of the equivalence when the model quadric is strongly pseudoconvex, recovering a result of Tumanov [14]. Our proof is based on the explicit expression of stationary discs, and opens up a conjecture for the unique 1-jet determination to hold when the model is not necessarily strongly pseudoconvex.

Normal form for pseudo-Einstein contact forms and intrinsic CR normal coordinates

We give a normal form for pseudo-Einstein contact forms and apply it to construct intrinsic CR normal coordinates parametrized by the structure group of CR geometry. The proof is based on the construction of parabolic normal coordinates by Jerison and Lee.

CR embeddings of CR manifolds

We improve results of Baouendi, Rothschild and Treves and of Hill and Nacinovich by finding a much weaker sufficient condition for a CR manifold of type (n, k) to admit a local CR embedding into a CR manifold of type (n+ell ,k-ell ). While their results require the existence of a finite dimensional solvable transverse Lie algebra of vector fields, we require only a finite dimensional extension.

On Degenerate Para-CR Structures: Cartan Reduction and Homogeneous Models

Motivated by recent works in Levi degenerate CR geometry, this article endeavors to study the wider and more flexible para-CR structures for which the constraint of invariancy under complex conjugation is relaxed. We consider 5-dimensional para-CR structures whose Levi forms are of constant rank 1 and that are 2-nondegenerate both with respect to parameters and to variables. Eliminating parameters, such structures may be represented modulo point transformations by pairs of PDEs zy = F(x,y,z,zx) & zxxx = H(x,y,z,zx,zxx), with F independent of zxx and $F_{z_{x}z_{x}} \neq 0$ , that are completely integrable ${D_{x}^{3}} F = {\Delta }_{y} H$ , Performing at an advanced level Cartan’s method of equivalence, we determine all concerned homogeneous models, together with their symmetries:

A high-temperature adhesive for stainless steel 304: Effect of Cr growth at the interface of alkali silicate adhesive

With the recent advancements in the aerospace industry, the demand for adhesive materials with resistance to high temperatures has increased. Although the existing organic adhesive materials have a high adhesion strength, their heat resistance and pyrolysis temperature are low, thus limiting their application under high-temperature conditions. Therefore, we attempted to bond stainless steel 304 using an alkali silicate, an inorganic material-based adhesive with high heat resistance but low adhesive strength and analyzed the effects of surface oxidation of stainless steel 304 on its adhesion strength. The surface treatment of SUS304 was performed using mechanical polishing (MP) and electropolishing (EP). A Cr 2 O 3 dominant layer (CODL) and a Fe 2 O 3 dominant layer (FODL) were formed by heat treatment. The adhesion failed in MP, EP, and CODL specimens, when bonding was performed using mixed silicate (MS) because of the differences in thermal expansion coefficients between the metal and glass. FODL and CODL + FODL specimens displayed MS and adhesion strengths of 8.45 MPa and 11.36 MPa, respectively. Each surface and adhesion interface was analyzed through X-ray diffraction (XRD), cross-section scanning electron microscopy (SEM), and energy-dispersive X-ray spectroscopy (EDS), before and after adhesion. When CODL and FODL specimens were placed on both sides to bond with the MS, a needle-like Cr structure appeared on the MS layer from the Cr 2 O 3 layer that enhanced the mechanical adhesion with the MS and the overall adhesive strength. Based on these findings, we conclude that the adhesion strength can be improved by combining a CODL specimen and a FODL specimen and bonding with glass.

Maximal Dimension of Groups of Symmetries of Homogeneous 2-nondegenerate CR Structures of Hypersurface Type with a 1-dimensional Levi Kernel

We prove that for every n ≥ 3 the sharp upper bound for the dimension of the symmetry groups of homogeneous, 2-nondegenerate, (2n + 1)-dimensional CR manifolds of hypersurface type with a 1-dimensional Levi kernel is equal to n2 + 7, and simultaneously establish the same result for a more general class of structures characterized by weakening the homogeneity condition. This supports Beloshapka’s conjecture stating that hypersurface models with a maximal finite dimensional group of symmetries for a given dimension of the underlying manifold are Levi nondegenerate.