CR Manifolds Research Articles

Zero‐curvature subconformal structures and dispersionless integrability in dimension five

AbstractWe extend the recent paradigm “Integrability via Geometry” from dimensions 3 and 4 to higher dimensions, relating dispersionless integrability of partial differential equations to curvature constraints of the background geometry. We observe that in higher dimensions on any solution manifold, the symbol defines a vector distribution equipped with a subconformal structure, and the integrability imposes a certain compatibility between them. In dimension 5, we express dispersionless integrability via the vanishing of a certain curvature of this subconformal structure. We also obtain a “master equation” governing all second‐order dispersionless integrable equations in 5D, and count their functional dimension. It turns out that the obtained background geometry is parabolic of the type . We provide its Cartan‐theoretic description and compute the harmonic curvature components via the Kostant theorem. Then, we relate it to 3D projective and 4D conformal geometries via twistor theory, discuss symmetry reductions and nested Lax sequences, as well as give another interpretation of dispersionless integrability in 5D through Levi‐degenerate CR structures in 7D.

Chen's Inequality for CR-warped Products in Locally Metallic Riemannian Manifolds

In this paper, we study CR warped product submanifolds in locally metallic Riemannian manifolds. We provide several non-trivial examples of such submanifolds. We establish a sharp inequality known as Chen's inequality for the squared norm of the second fundamental form. We also discuss the equality case of Chen's inequality.

Twistor and Reflector Spaces for Paraquaternionic Contact Manifolds

We consider certain fiber bundles over paraquaternionic contact manifolds, called twistor and reflector spaces. We show that the twistor space carries an integrable CR structure (Cauchy–Riemann structure) and the reflector space is an integrable para-CR structure, both with neutral signatures.

Existence results for the Einstein-scalar field Lichnerowicz equations on CR manifold in the null case

Existence results for the Einstein-scalar field Lichnerowicz equations on CR manifold in the null case

Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds

Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds

Hydrogen evolution reaction performance of Pt modified monolayer MC2 MXene (M=W, Cr, Mo)

The development of cost-effective hydrogen evolution reaction (HER) electrocatalysts has become pivotal in hydrogen production field. Because of the remarkable catalytic performance of Pt metal, developing low proportion Pt-based electrocatalysts should be very meaningful for entrepreneurs and scientists. In order to further decrease the economic cost, the HER electrocatalytic performance of low metal ratio Pt-doped MC2(M=Mo, W, and Cr) structures is investigated by first-principles calculations. It merits our attention that in monolayer WC2 when a Pt atom replaces a W atom, where HER occurs, the structure has a lowest overpotential of only 0.03 V, which is much lower than the reported value of pure Pt (111) (0.09 V) and the lowest value (-0.31 V) of WCx(x=1–4). It can be observed that some energy bands of Pt-WC2 cross the Fermi level, resulting in remarkable electronic conductivity. Specifically, Pt-WC2 shows good mechanical, dynamic, and thermal stabilities. Thus, single Pt modified monolayer WC2 can serve as an efficient economic electrocatalyst for HER. These findings can support strong theoretical basis for the experimental researches to develop a kind of excellent quality and reasonable price catalyst in the future.

Models of CR Manifolds and Their Symmetry Algebras

Models of CR Manifolds and Their Symmetry Algebras

Hardy Spaces and Canonical Kernels on Quadric CR Manifolds

CR functions on an embedded quadric M always extend holomorphically to M+iΓM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M+i\\Gamma _M$$\\end{document} where ΓM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _M$$\\end{document} is the closure of the convex hull of the image of the Levi form. When ΓM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _M$$\\end{document} is a closed polygonal cone, we show that the Bergman kernel on the interior of M+iΓM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M+i\\Gamma _M$$\\end{document} is a derivative of the Szegö kernel. Moreover, we develop the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} Hardy space theory which turns out to be particularly robust. We provide examples that show that it is unclear how to formulate a corresponding relationship between the Bergman and Szegö kernels on a wider class of quadrics.

On the variation of the Einstein–Hilbert action in pseudohermitian geometry

Abstract In this paper, we compute the first and second variation of the normalized Einstein–Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres. While the situation is quite similar to the Riemannian case in dimension greater than or equal to five, in three dimensions, we observe a crucial difference, which mainly depends on the embeddable character of the perturbed CR structure.

CR Compactification for Asymptotically Locally Complex Hyperbolic Almost Hermitian Manifolds

In this article, we consider a complete, non-compact almost Hermitian manifold whose curvature is asymptotic to that of the complex hyperbolic space. Under natural geometric conditions, we show that such a manifold arises as the interior of a compact almost complex manifold whose boundary is a strictly pseudoconvex CR manifold. Moreover, the geometric structure of the boundary can be recovered by analysing the expansion of the metric near infinity.

A characterization of homogeneous three-dimensional CR manifolds

We characterize homogeneous three-dimensional CR manifolds, in particular Rossi spheres, as critical points of a certain energy functional that depends on the Webster curvature and torsion of the pseudohermitian structure.

Optimal decay for solutions of nonlocal semilinear equations with critical exponent in homogeneous groups

In this paper, we establish the sharp asymptotic decay of positive solutions of the Yamabe type equation $\mathcal {L}_s u=u^{\frac {Q+2s}{Q-2s}}$ in a homogeneous Lie group, where $\mathcal {L}_s$ represents a suitable pseudodifferential operator modelled on a class of nonlocal operators arising in conformal CR geometry.

Quasi Yamabe Solitons on CR Submanifolds of Maximal CR Dimension in Kähler Manifolds

In this paper we give necessary and sufficient conditions for a CR submanifold of maximal CR dimension in arbitrary Kähler manifold to admit (quasi-)Yamabe structure, with naturally chosen soliton vector field. When the ambient manifold is a non-flat complex space form, we give a complete classification of such solitons, under certain conditions.

Asymptotic strictly pseudoconvex CR structure for asymptotically locally complex hyperbolic manifolds

AbstractIn this paper, we build a compactification by a strictly pseudoconvex CR structure for a complete and non-compact Kähler manifold whose curvature tensor is asymptotic to that of the complex hyperbolic space. To do so, we study in depth the evolution of various geometric objects that are defined on the leaves of some foliation of the complement of a suitable convex subset, called an essential subset, whose leaves are the equidistant hypersurfaces above this latter subset. With a suitable renormalization which is closely related to the anisotropic nature of the ambient geometry, the above mentioned geometric objects converge near infinity, inducing the claimed structure on the boundary at infinity.

A novel ionic liquid-entrapped MIL-101(Cr) framework with enhanced removal efficiency towards phosphate from aqueous solution.

Developing adsorbent materials with high adsorptive dephosphorization (ADP) is significant for treating phosphate from aqueous solutions and eutrophic water. Herein, the MIL-101(Cr) framework was entrapped ionic liquid (IL) of 1-butyl-3-methylimidazoliumbromide ionic liquid ([C4mem]+[Br]-) using a ship-in-a-bottle approach to obtain novel adsorbents [C4mem]+[Br]-@MIL-101(Cr) contained varied IL contents, namely C4mem@MIL-101. The characterization results revealed that the formed [C4mem]+[Br]- molecules interacted with the MIL-101(Cr) frameworks, enhanced their stability, and offered additional adsorption sites. The batch adsorptions of phosphate showed that the optimized C4mem@MIL-101 adsorbent loaded with ~ 7% IL-based N content had the highest phosphate absorbing capacity of ~ 200mg/g, outperforming the pristine MIL-101(Cr) and other adsorbents. The ADP efficiency was facilitated in the acidic media, where the phosphate ions of H2PO4- and HPO42- captured onto the C4mem@MIL-101 via several interactions, including electrostatic attraction, H-bonds, and chemical interactions. In the meantime, the coexisting anions diminished the phosphate adsorption because they competed with the pollutants at adsorption sites. Furthermore, phosphate treatment under the continuous fixed-bed conditions showed that 1g of the polyvinyl alcohol (PVA)-mixed C4mem@MIL-101 pellets purified 25l of water containing phosphate with a 1mg/l concentration. The results suggest that the novel [C4mem]+[Br]-@MIL-101(Cr) structure had a high potential for treating phosphate in aqueous solutions.

Almost CR manifolds with contracting CR automorphism

Almost CR manifolds with contracting CR automorphism

Hypoelliptic Operators in Geometry

The workshop titled Hypoelliptic Operators in Geometry was coorganized by Davide Barilari (Padova), Xiaonan Ma (Paris), Nikhil Savale (K¨oln) and Yi Wang (Baltimore). It was well attended by 55 participants, with 45 of them being present in person and 10 being online. The participants came from several continents, age groups and included male as well as female researchers. Several interesting themes were discussed including: analysis around Kohn’s Laplacian in CR geometry, analogous covariant operators arising in conformal geometry, the spectral theory of the sub-Riemannian Laplacian, pseudodifferential calculi in non-commutative geometry and the geometric applications of Bismut’s hypoelliptic Laplacians.

Analysis of the critical CR GJMS operator

Abstract: The critical CR GJMS operator on a strictly pseudoconvex CR manifold is a non-hypoelliptic CR invariant differential operator. We prove that, under the embeddability assumption, it is essentially self-adjoint and has closed range. Moreover, its spectrum is discrete, and the eigenspace corresponding to each non-zero eigenvalue is a finite-dimensional subspace of the space of smooth functions. As an application, we obtain a necessary and sufficient condition for the existence of a contact form with zero CR $Q$-curvature.

Commutativity of quantization with conic reduction for torus actions on compact CR manifolds

We define conic reductions Xνred\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{\ extrm{red}}_{\ u }$$\\end{document} for torus actions on the boundary X of a strictly pseudo-convex domain and for a given weight ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} labeling a unitary irreducible representation. There is a natural residual circle action on Xνred\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{\ extrm{red}}_{\ u }$$\\end{document}. We have two natural decompositions of the corresponding Hardy spaces H(X) and H(Xνred)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(X^{\ extrm{red}}_{\ u })$$\\end{document}. The first one is given by the ladder of isotypes H(X)kν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(X)_{k\ u }$$\\end{document}, k∈Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in {\\mathbb {Z}}$$\\end{document}; the second one is given by the k-th Fourier components H(Xνred)k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(X^{\ extrm{red}}_{\ u })_k$$\\end{document} induced by the residual circle action. The aim of this paper is to prove that they are isomorphic for k sufficiently large. The result is given for spaces of (0, q)-forms with L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-coefficient when X is a CR manifold with non-degenerate Levi form.

Homogeneous CR and Para-CR Structures in Dimensions 5 and 3

Homogeneous CR and Para-CR Structures in Dimensions 5 and 3