CR Manifolds Research Articles

Some Remarks on the Total CR Q and Q'-Curvatures

We prove that the total CR $Q$-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the $P^\prime$-operator and the CR invariance of the total $Q^\prime$-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.

Non locally trivializable CR line bundles over compact Lorentzian CR manifolds

We consider compact CR manifolds of arbitrary CR codimension that satisfy certain geometric conditions in terms of their Levi form. Over these compact CR manifolds, we construct a deformation of the trivial CR line bundle over M which is topologically trivial over M but fails to be even locally CR trivializable over any open subset of M. In particular, our results apply to compact Lorentzian CR manifolds of hypersurface type.

On the analyticity of CR-diffeomorphisms

In any positive CR-dimension and CR-codimension we provide a construction of real-analytic embedded CR-structures, which are $C^\infty$ CR-equivalent, but are inequivalent holomorphically.

Lower extremities for generalized normalized δ-Casorati curvatures of bi-slant submanifolds in generalized complex space forms

In this paper, we obtain the inequalities for the generalized normalized δ-Casorati curvature and the normalized scalar curvature for different submanifolds in generalized complex space form, which is based on an optimization procedure involving a quadratic polynomial in the components of the second fundamental form and characterizes the submanifolds on which equalities hold. We also develop, the same inequalities for semi-slant, hemi-slant, CR, slant, invariant and anti-invariant submanifolds in the same target space and consider the equality case. Moreover, we obtain a geometric inequality involving Casorati curvature for warped product bi-slant submanifolds in same ambient and obtain an obstruction result.

Szegő kernel asymptotics and Morse inequalities on CR manifolds with $S^1$ action

Let $X$ be a compact connected CR manifold of dimension $2n-1, n\geq 2$. We assume that there is a transversal CR locally free $S^1$ action on $X$. Let $L^k$ be the $k$-th power of a rigid CR line bundle $L$ over $X$. Without any assumption on the Levi-form of $X$, we obtain a scaling upper-bound for the partial Szegő kernel on $(0,q)$-forms with values in $L^k$. After integration, this gives the weak Morse inequalities. By a refined spectral analysis, we also obtain the strong Morse inequalities in CR setting. We apply the strong Morse inequalities to show that the Grauert-Riemenschneider criterion is also true in the CR setting.

A family of compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^2$ without umbilical points

We prove the following: For $\epsilon>0$, let $D_\epsilon$ be the bounded strictly pseudoconvex domain in $\mathbb C^2$ given by \begin{equation*} (\log|z|)^2+(\log|w|)^2<\epsilon^2. \end{equation*} The boundary $M_\epsilon:=\partial D_\epsilon\subset \mathbb C^2$ is a compact strictly pseudoconvex CR manifold without umbilical points. This resolves a long-standing question in complex analysis that goes back to the work of S.-S. Chern and J. K. Moser in 1974.

The Log Term in the Bergman and Szegő Kernels in Strictly Pseudoconvex Domains in $\mathbb C^2$

In this paper, we consider bounded strictly pseudoconvex domains D\subset \mathbb C^2 with smooth boundary M=M^3:=\partial D , and the asymptotic expansion of the Bergman kernel on the diagonal K_B\sim \frac{\phi_{B}}{\rho^{n+1}}+\psi_B\log\rho, where \rho&gt;0 is a Fefferman defining equation for D . The Ramadanov Conjecture states that if the log term \psi_B vanishes to infinite order on M , then M is locally spherical. In \mathbb C^2 , the validity of this conjecture is known and follows from work of Boutet de Monvel, Burns, and Graham; indeed, it suffices that \psi_B=O(\rho^2) locally on M to conclude that M is locally spherical. On the other hand, it is also known that the boundary values alone of the log term b\psi_B:=(\psi_B)|_M on M does not determine the CR geometry of M locally; e.g., the vanishing of b\psi_B on an open subset of M does not imply that M is locally spherical there. The main result in this paper, however, is that if D\subset \mathbb C^2 is assumed to have transverse symmetry, then the global vanishing of b\psi_B on M implies that M is locally spherical. A similar result is proved for the Szegő kernel.

Sums of CR functions from competing CR structures

In this paper we characterize sums of CR functions from competing CR structures in two scenarios. In one scenario the structures are conjugate and we are adding to the theory of pluriharmonic boundary values. In the second scenario the structures are related by projective duality considerations. In both cases we provide explicit vector field-based characterizations for two-dimensional circular domains satisfying natural convexity conditions.

An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group

We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is proved in terms of p-area which naturally arises from the variation of volume. The application makes a connection between CR geometry and integral geometry.

Convergence of the CR Yamabe flow

We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold M of real dimension $$2n+1$$ . We prove convergence of the CR Yamabe flow when $$n=1$$ or M is spherical.

Structural resemblance and variation in transition metal complexes derived from dipicolinic acid and guanylhydrazine

New aminoguanidinium transition metal dipicolinates of the formula (AgunH)[MIII(dip)2]·xH2O, M = Cr, x = 1.5; Fe, x = 2, (AgunH)2[MII(dip)2]·nH2O, where M = Mn, n = 2; Co or Ni, n = 0 and Zn, n = 1, and [Cu(Agun)2][Cu3(dip)4(Agun)2]·4H2O have been prepared by the reaction of respective metal nitrates with aqueous mixture of dipicolinic acid (H2dip) and aminoguanidine bicarbonate (AgunH·HCO3) in appropriate mole ratios. These compounds have been characterized by various physico-chemical techniques. Among them Ni, Cu, Cr and Fe complexes have been structurally characterized. The asymmetric units of the Ni and Cr structures contain two independent molecules, whereas Fe has four. Unlike these, the copper compound has a unique structure with a tri-nuclear centrosymmetric anionic complex and cationic bis(aminoguanidine)copper(II). In all of these compounds, the dipicolinate dianion is acting as a tridentate (O,N,O) donor ligand, except copper in-which one of the dipicolinates is tetradentate (O,N,O,O). Cytotoxicity and DPPH scavenging activity studies reveal that the copper complex exhibits superior activity to that of cobalt and nickel.

Spinorially Twisted Spin Structures, III: CR Structures

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 almost CR structure in our spinorial setup, including the classical integrability of a CR structure as well as those implied by Killing-type conditions on the partially pure spinor field. In the codimension one case, we develop a spinorial description of strictly pseudoconvex CR manifolds, metric contact manifolds, and Sasakian manifolds. Finally, we study hypersurfaces of Kahler manifolds via partially pure Spin $$^c$$ spinors.

Levi–Kähler reduction of CR structures, products of spheres, and toric geometry

We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of such quotients, and find Levi-Kahler quotients of products of 3-spheres which are extremal in a weighted sense introduced by G. Maschler and the first author.

On the phase evolution of arc evaporated Al-Cr-based intermetallics and oxides

(Al,Cr)2O3 solid solutions in corundum-structure are of great potential for protective applications facing severe mechanical and thermal conditions. In order to elucidate the influence of cathode composition and reactive gas flow on the phase formation of (Al1‐xCrx)2O3, Al0.75Cr0.25, Al0.70Cr0.30, Al0.50Cr0.50, and Al0.25Cr0.75 cathodes were utilised to synthesise coatings with and without oxygen reactive gas. Transmission electron microscopy studies reveal successive microstructural modifications from intermetallic coatings towards oxides with M2O3 stoichiometry. The transition from substoichiometric (suggesting intermetallic γ2-Al8Cr5, AlCr2, and/or Cr structures with distorted lattices due to oxygen incorporation) to stoichiometric oxides is connected with the formation of metastable cubic (Al,Cr)2O3 phases for Al-rich compositions. With decreasing Al-content of the cathodes, the critical oxygen flow for this transformation increases (i.e., 380sccm for Al0.75Cr0.25, 400sccm for Al0.70Cr0.30, and 420sccm for Al0.50Cr0.50). Only when using the high-Cr containing Al0.25Cr0.75 cathode, the coatings directly crystallised with corundum-type (Al,Cr)2O3-solid solutions for O2 flow rates above ~570sccm.

Effect of surface tension and geometry on cavitation in soft solids

Unbounded growth of cavity in soft solids subjected to internal pressure is a commonly observed phenomenon. Such phenomenon has been harnessed by developing cavitation rheology technique (CR) to investigate the local mechanical properties of many complex soft materials. The elasticity of the material, surface energy, and geometric factors in combination can dictate the cavitation behavior in a complex manner. We report a finite element based framework capturing the coupled effect of elastic strain energy and surface tension on the cavitation phenomenon. We show results for a spherical cavity in an infinite elastic media and for the CR geometry. The surface tension is shown to increase the critical pressure for cavitation. The energy release rate also depends on the surface tension. In CR geometry, by varying the distance between the needle and the immovable sample boundaries, we have captured the conditions leading to confinement. Our results provide new understanding regarding the effects of geometry and surface tension on the cavitation phenomenon in soft solids.

On the stability of equivariant embedding of compact CR manifolds with circle action

We prove the stability of the equivariant embedding of compact strictly pseudoconvex CR manifolds with transversal CR circle action under circle invariant perturbations of the CR structures.

Bach-Flat Kähler Surfaces

A Riemannian metric on a compact 4-manifold is said to be Bach-flat if it is a critical point for the $$L^2$$ -norm of the Weyl curvature. When the Riemannian 4-manifold in question is a Kahler surface, we provide a rough classification of solutions, followed by detailed results regarding each case in the classification. The most mysterious case prominently involves 3-dimensional CR manifolds.

Paramagnetism in Metallacarboranes: The Polyhedral Chromadicarbaborane Systems.

The chromadicarbaboranes CpCrC2Bn-3Hn-1 (8 ≤ n ≤ 12) are of interest in providing stable paramagnetic deltahedral metallaboranes among which the 12-vertex CpCrC2B9H11 has been synthesized by Hawthorne and co-workers. Density functional theory shows that the lowest-energy such structures are quartet spin-state Cr(III) structures in which the central CrC2Bn-3 units exhibit most spherical closo deltahedral geometries similar to those found in the borane dianions BnHn2-. Higher-energy doublet CpCrC2Bn-3Hn-1 (8 ≤ n ≤ 11) structures are found exhibiting central CrC2Bn-3 isocloso deltahedral geometries, thereby providing a degree 6 vertex for the chromium atom. The lowest-energy CpCrC2Bn-3Hn-1 (8 ≤ n ≤ 11) structures all have both carbon atoms at degree 4 vertices. However, the lowest-energy CpCrC2B9H11 structures all have central CrC2B9 icosahedra and thus lack degree 4 vertices for the carbon atoms. For all of the CpCrC2Bn-3Hn-1 (8 ≤ n ≤ 12) systems the lowest-energy isomers are those with the maximum number of Cr-C edges in contrast to the related CpCoC2Bn-3Hn-1 systems.

On the singularities of the Szegő projections on lower energy forms

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$-forms. We show that the spectral function of $\Box^{(q)}_b$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of $\Box^{(q)}_b$ in $]0,\infty[$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szeg\"o kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szeg\"o kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR $S^1$ actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR $S^1$ action can be CR embedded into $\mathbb{C}^N$, for some $N\in\mathbb N$.

Lorentzian CR structures and nonembeddability

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.