CR Manifolds Research Articles

Key Cr species controlling Cr stability in contaminated soils before and chemical stabilization at a remediation engineering site

Linking chromium (Cr) speciation with its stability in soils is vital because insoluble Cr(VI) and chemically adsorbed Cr(VI) could hinder the remediation efficiency and release Cr(VI) for a prolonged period of time. In this study, we investigated key Cr species to probe the mechanisms controlling the release of insoluble Cr(VI) at Cr-contaminated sites using synchrotron-based X-ray absorption near-edge structure (XANES) for the first time. Chromite, stichtite and Cr-silicate were predominant forms of Cr(III). Insoluble Cr(VI) was hosted by layered double hydroxides (LDHs) such as brownmilerite and hydrotalcite. Anion competition tests documented a substitution of absorbed Cr(VI) by SO42- and NO3-. Acid extraction released 6.7–25.7% more Cr(VI) than anion extraction, possibly attributing to the erosion of LDH and CaCrO4 in calcite rather than Cr-bearing minerals. Brown and red soils released maximally 62% and 44% of total Cr(VI) by 10 mol/(kg soil) and 2 mol/(kg soil) of H+, respectively. SO42-, H2O and H+ contributed to more release of total Cr(VI) in brown soils (22%, 33% and 7%) than red soils (25%, 17% and 2%). More crystalline Cr structures were found after chemical stabilization, indicating a higher Cr stability in chemically stabilized soils. Cr and Mn exhibited an overlapped distribution pattern in both contaminated and chemically stabilized soils, hinting at the re-oxidation of Cr(III). Insoluble Cr(VI) could be released by acidic rainfalls and soil organic matters, posing potential threats to Cr long-term stability in field-scale remediation.

Obstruction flat rigidity of the CR 3-sphere

Abstract We consider the obstruction flatness problem for small deformations of the standard CR 3-sphere. That rigidity holds for the CR sphere was previously known (in all dimensions) for the case of embeddable CR structures, where it also holds at the infinitesimal level. In the 3-dimensional case, however, a CR structure need not be embeddable. Unlike in the embeddable case, it turns out that in the nonembeddable case there is an infinite-dimensional space of solutions to the linearized obstruction flatness equation on the standard CR 3-sphere and this space defines a natural complement to the tangent space of the embeddable deformations. In spite of this, we show that the CR 3-sphere does not admit nontrivial obstruction flat deformations, embeddable or nonembeddable.

On Nirenberg's non-embeddable CR structure

We study -valued CR functions of Theodoresco class on the Heisenberg group equipped with the non-embeddable CR structure discovered by Nirenberg [On a question of Hans Lewy. Russian Math Surveys. 1974;29:251–262] where is an arbitrary complex Fréchet space. If is Nirenberg's perturbation of Lewy's operator , we show that for every open neighbourhood of the origin, no two solutions , , of Theodoresco class , to the tangential Cauchy–Riemann equations are functionally independent.

Concentration phenomena for fractional CR Yamabe type flows

In this paper, we consider fractional CR Yamabe type equations of order 2γ, 0<γ<n+1, on the standard CR Manifold (S2n+1,θ0), n≥1.By studying the asymptotic behaviors of the associated pseudo-gradient flows, we give a complete description of the bubbling phenomena of the related variational structure. We were able to fully characterize concentration phenomena for all blow-up points. Our results are novel and require a subtle analysis. All previous contributions addressed this aspect in a perturbative setting or only in the three-dimensional case. We also provide a variety of functions which can be achieved as conformal fractional CR Q-curvatures on S2n+1.

On CR singular CR images

We say that a CR singular submanifold [Formula: see text] has a removable CR singularity if the CR structure at the CR points of [Formula: see text] extends through the singularity as an abstract CR structure on [Formula: see text]. We study such real-analytic submanifolds, in which case removability is equivalent to [Formula: see text] being the image of a generic real-analytic submanifold [Formula: see text] under a holomorphic map that is a diffeomorphism of [Formula: see text] onto [Formula: see text], what we call a CR image. We study the stability of the CR singularity under perturbation, the associated quadratic invariants, and conditions for removability of a CR singularity. A lemma is also proved about perturbing away the zeros of holomorphic functions on CR submanifolds, which could be of independent interest.

Twisting non-shearing congruences of null geodesics, almost CR structures and Einstein metrics in even dimensions

We investigate the geometry of a twisting non-shearing congruence of null geodesics on a conformal manifold of even dimension greater than four and Lorentzian signature. We give a necessary and sufficient condition on the Weyl tensor for the twist to induce an almost Robinson structure, that is, the screen bundle of the congruence is equipped with a bundle complex structure. In this case, the (local) leaf space of the congruence acquires a partially integrable contact almost CR structure of positive definite signature. We give further curvature conditions for the integrability of the almost Robinson structure and the almost CR structure, and for the flatness of the latter. We show that under a mild natural assumption on the Weyl tensor, any metric in the conformal class that is a solution to the Einstein field equations determines an almost CR-Einstein structure on the leaf space of the congruence. These metrics depend on three parameters, and include the Fefferman-Einstein metric and Taub-NUT-(A)dS metric in the integrable case. In the non-integrable case, we obtain new solutions to the Einstein field equations, which, we show, can be constructed from strictly almost Kaehler-Einstein manifolds.

Schatten-von Neumann classes of integral operators

In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in Rn of finite measure, and conditions for operators on Rn given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups.

Structure of cognitive disorders and predictors of their occurrence in patients after cardiac surgery

Objective — to determine the presence and severity of moderate cognitive disorders (CD) in patients before and after cardiac surgery in the early postoperative period&#x0D; Methods and subjects. 56 patients were examined, including 19 (33.9 %) men and 37 (66,1 %) women , mean age 60.86 ± 8.86. Cardiac surgery was performed for ischemic heart disease in 37 (66.1 %) patients, for valvular heart disease — in 19 (33.9 %, p = 0.02). Assessment of cognitive functions was performed before surgery, on the 3rd and 7th day of the postoperative period using the Montreal Cognitive Test.&#x0D; Results. Before surgery, most patients (75.0 %) had mild CRs, 5.4 % had moderate CRs (p &lt;0.01), and 19.6 % had no CRs (p = 0.02). In general, the distribution by the severity of CR in the early postoperative period did not differ from that before surgery, but there were patients with severe CR (3.6 and 1.8 %, respectively, on the 3rd and 7th days, p = 0.05) ...In the structure of CD, on the 3rd day of the postoperative period, there was a significant decrease in visually constructive skills (4.07 versus 3.7; p &lt; 0.001), a deterioration in the ability to consistently calculate (2.66 versus 2.45; p = 0.02), repetition phrases (1.16 versus 1.0; p = 0.02). On the 7th day of the postoperative period, there was a significant decrease in verbal speed (0.48 versus 0.32; p = 0.006) and an improvement in memory (1.79 versus 2.29; p = 0.01). Probable predictors of the onset of cognitive disorders in the early postoperative period are: a history of stroke — r = –0.282; p = 0.04. There was a correlation with the presence of atherosclerotic lesions of the coronary arteries according to coronary angiography (r = –0.259; p = 0.05), surgery performed under cardiopulmonary bypass (r = 0.29; p = 0.03), and a tendency towards statistical significance of correlation with the presence of signs of dyslipidemia according to lipid profile data (r = –0.227; p = 0.09).&#x0D; Conclusions. It was found that 75 % of patients had mild CR before surgery. The distribution according to the severity of CR in the early postoperative period differed from that before surgery by the appearance of patients with severe CR. In the CR structure, on the 3rd day of the postoperative period, a statistically significant decrease in visual‑constructive skills, a deterioration in the ability to consistently calculate and repeat a phrase were observed. On the 7th day, a statistically significant decrease in verbal speed and an improvement in memory were noted. Probable predictors of CR in the early postoperative period are a history of acute cerebrovascular accident, atherosclerotic lesion of the coronary arteries according to coronary angiography, surgery performed under cardiopulmonary bypass, the presence of signs of dyslipidemia according to lipid profile data.&#x0D;

Left-invariant CR structures on 3-dimensional Lie groups

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, mostly expository, we illustrate and clarify Cartan’s results and methods by providing detailed classification results in modern language for four 3-dimensional Lie groups. In particular, we find that \({\mathrm {SL}_2({\mathbb {R}})}\) admits two one-parameter families of left-invariant CR structures, called the elliptic and hyperbolic families, characterized by the incidence of the contact distribution with the null cone of the Killing metric. Low dimensional complex representations of \({\mathrm {SL}_2({\mathbb {R}})}\) provide CR embedding or immersions of these structures. The same methods apply to all other 3-dimensional Lie groups and are illustrated by descriptions of the left-invariant CR structures for \(\mathrm {SU}_2\), the Heisenberg group, and the Euclidean group.

Uniqueness results on CR manifolds

In this paper we show the uniqueness of the constant solution to a general Yamabe-type equation on a compact pseudohermitian manifold. The Riemannian version was first proved by Bidaut-V eron and V eron. In another direction, we make use of the vanishing property of two eigenfunctions of the sub-Laplacian to conclude the nonvanishing of the first cohomology group of a compact pseudohermitian manifold. The corresponding version for the Rieman- nian case was first proved by Gichev.

The Bochner-Kodaira-Nakano formula for line bundles on CR manifolds with transversal CR R-action

The Bochner-Kodaira-Nakano formula for line bundles on CR manifolds with transversal CR R-action

Smooth solutions to the complex plateau problem

The paper builds on work of Du, Gao, and Yau. The main result provides a characterization of smooth solutions, up to normalization, of the complex Plateau problem for strongly pseudoconvex Calabi–Yau CR manifolds of dimension $2n - 1 \geq 5$ and in the hypersurface case when $n = 2$. The latter case was completely solved by Yau for $n \geq 3$ but only partially solved by Du and Yau for $n = 2$. As an application, we determine the existence of a link-theoretic invariant of normal isolated singularities that distinguishes smooth points from singular ones.

Extension of CR structures on pseudoconvex CR-manifolds with comparable eigenvalues of the Levi-form

Let be a smoothly bounded orientable pseudoconvex CR manifold of finite 1-type with comparable eigenvalues of the Levi-form. Then we extend the given CR structure on to an integrable almost complex structure on the concave side of M when . If , we extend the structure on the convex side. Therefore we may regard M as a real hypersurface in when .

An optimization problem arising in CR geometry

We determine the asymptotic behavior as the degree tends to infinity of the minimal L1 norm m(n,d) of the solution of an optimization problem arising when studying polynomial sphere maps. Here n is the source dimension and d is the degree. We provide upper and lower bounds for m(n,d). We use these bounds to show that the function d→m(n,d) is monotone increasing in d. We prove that limd→∞m(n,d) d=n(n−1). Let N(n,d) denote the minimum possible target dimension of a monomial sphere map of degree d. We show, in source dimension unequal to 2, that limd→∞m(n,d) N(n,d)=n. The limit is 4 when n=2. We discuss some complicated results obtained by coding when n=2.

Sasakian geometry on sphere bundles

The purpose of this paper is to study the Sasakian geometry on odd dimensional sphere bundles over a smooth projective algebraic variety N with the ultimate, but probably unachievable goal of understanding the existence and non-existence of extremal and constant scalar curvature Sasaki metrics. We apply the fiber join construction of Yamazaki [48] for K-contact manifolds to the Sasaki case. This construction depends on the choice of d+1 integral Kähler classes [ωj] on N that are not necessarily colinear in the Kähler cone. We show that the colinear case is equivalent to a subclass of a different join construction originally described in [7,11], applied to the spherical case by the authors in [19,20] when d=1, and known as cone decomposable [13]. The non-colinear case gives rise to infinite families of new inequivalent cone indecomposable Sasaki contact CR structures on certain sphere bundles. We prove that the Sasaki cone for some of these structures contains an open set of extremal Sasaki metrics and, for certain specialized cases, the regular ray within this cone is shown to have constant scalar curvature. We also compute the cohomology groups of all such sphere bundles over a product of Riemann surfaces.

Three-Dimensional CR Submanifolds in $S^6(1)$ with Umbilical Direction Normal to $\mathcal{D}_3$

It is well known that the sphere $S^6(1)$ admits an almost complex structure $J$ which is nearly K\"{a}hler. A submanifold $M$ of an almost Hermitian manifold is called a CR submanifold if it admits a differentiable almost complex distribution $\mathcal{D}_1$ such that its orthogonal complement is a totally real distribution. In this case the normal bundle of the submanifold also splits into two distributions $\mathcal{D}_3$, which is almost complex, and a totally real complement. In the case of the proper three-dimensional CR submanifold of a six-dimensional manifold the distribution $\mathcal{D}_3$ is non-trivial. Here, we investigate three-dimensional CR submanifolds of the sphere $S^6(1)$ admitting an umbilic direction orthogonal to $\mathcal{D}_3$ and show that such submanifolds do not exist.

A CR singular analogue of Severi’s theorem

Real-analytic CR functions on real-analytic CR singular submanifolds are not in general restrictions of holomorphic functions, unlike in the CR nonsingular case. We give a simple condition that completely characterizes those quadric CR singular manifolds of codimension 2 in ${\mathbb C}^{n+1}$ for which an extension result holds. Consequently, we obtain an extension result for general real-analytic CR singular submanifolds of codimension 2. As applications we give a condition for the flattening of such submanifolds, and we classify CR singular images of CR submanifolds up to second order.

Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation

Abstract This article is devoted to the local geometry of everywhere 2-nondegenerate CR manifolds M of hypersurface type. An absolute parallelism for such structures was recently constructed independently by Isaev and Zaitsev, Medori and Spiro, and Pocchiola in the minimal possible dimension ( dim ⁡ M = 5 {\dim M=5} ), and for dim ⁡ M = 7 {\dim M=7} in certain cases by the first author. In the present paper, we develop a bigraded (i.e., ℤ × ℤ {\mathbb{Z}\times\mathbb{Z}} -graded) analog of Tanaka’s prolongation procedure to construct an absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol – a complex bigraded vector space – containing all essential information about the CR structure. Under the additional regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation, endowed with an anti-linear involution, and prove that for any CR structure with a given regular symbol there exists a canonical absolute parallelism on a bundle whose dimension is that of the bigraded universal algebraic prolongation. Moreover, we show that for each regular symbol there is a unique (up to local equivalence) such CR structure whose algebra of infinitesimal symmetries has maximal possible dimension, and the latter algebra is isomorphic to the real part of the bigraded universal algebraic prolongation of the symbol. In the case of 1-dimensional Levi kernel we classify all regular symbols and calculate their bigraded universal algebraic prolongations. In this case, the regular symbols can be subdivided into nilpotent, strongly non-nilpotent, and weakly non-nilpotent. The bigraded universal algebraic prolongation of strongly non-nilpotent regular symbols is isomorphic to the complex orthogonal algebra 𝔰 ⁢ 𝔬 ⁢ ( m , ℂ ) {\mathfrak{so}(m,\mathbb{C})} , where m = 1 2 ⁢ ( dim ⁡ M + 5 ) {m=\tfrac{1}{2}(\dim M+5)} . Any real form of this algebra – except 𝔰 ⁢ 𝔬 ⁢ ( m ) {\mathfrak{so}(m)} and 𝔰 ⁢ 𝔬 ⁢ ( m - 1 , 1 ) {\mathfrak{so}(m-1,1)} – corresponds to the real part of the bigraded universal algebraic prolongation of exactly one strongly non-nilpotent regular CR symbol. However, for a fixed dim ⁡ M ≥ 7 {\dim M\geq 7} the dimension of the bigraded universal algebraic prolongations of all possible regular CR symbols achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is 1 4 ⁢ ( dim ⁡ M - 1 ) 2 + 7 {\frac{1}{4}(\dim M-1)^{2}+7} .

Equivariant embeddings of strongly pseudoconvex Cauchy–Riemann manifolds

Let X be a CR manifold with transversal, proper CR action of a Lie group G. We show that the quotient X/G is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorizes uniquely over a holomorphic map on X/G. We then use this result and complex geometry to prove an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal G rtimes S^1-action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.

On equivalence problem for 2–nondegenerate CR geometries with simple models

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 models with a real simple CR automorphism group. In particular, we construct (local) embeddings of these models into complex space.