CR Manifolds Research Articles

Local control of 2-link robotic worms based on additional symmetries

On a new class of planar mechanisms, we demonstrate local control based on the symmetries of additional geometric structures. Using the example of a two-line robotic worm, we show that an additional CR structure on the control distribution allows local control of the chosen mechanism using symmetries.

Metal-organic frameworks-induced Self-Poling effect of polyvinylidene fluoride nanofibers for performance enhancement of triboelectric nanogenerator

Metal-organic frameworks (MOFs) exhibit various unique properties, but these properties have not been fully utilized for the performance enhancement of triboelectric nanogenerator (TENG). The MOFs with MIL-101(Cr) structure is a nanomaterial designed for extremely large pore structures than conventional MOF structures to absorb and desorb various solvents, which induces a new phenomenon, self-polling effect (SPE) in the organic materials. We verified and suggested the SPE mechanism of the N, N-dimethylformamide (DMF) solvent by the MIL-101(Cr) structured MOFs in the polyvinylidene fluoride (PVDF) nanofibers. Because of the SPE, the MOFs-based composite NFs showed enhanced TENG output performances by additionally induced β-phase in the PVDF NFs even after the electrospinning process. The composite PVDF NFs with optimal MOF content of 0.8 wt% showed an output power density of 568.8 µW/cm2, which is larger by 8.96 times compared to the pure PVDF NFs. After the SPE process, the output power density increased to 871.2 µW/cm2, which is 13.7 times larger than that of pure PVDF NFs. Different from the pure PVDF NFs, the MOFs-based composite PVDF NFs showed improved output powers after the SPE process regardless of MOF contents. Based on excellent performance, the TENG with the MOFs-based composite PVDF NFs was sufficiently utilized as a self-powered triggering sensor for smart home devices that can wirelessly operate various electronic devices.

Remarks on star products for non-compact CR manifolds with non-degenerate Levi form

Remarks on star products for non-compact CR manifolds with non-degenerate Levi form

Unique jet determination of CR maps into Nash sets

Let M⊂CN be a real-analytic CR submanifold, M′⊂CN′ a Nash set and EM′ the set of points in M′ of D'Angelo infinite type. We show that if M is minimal, then, for every point p∈M, and for every pair of germs of C∞-smooth CR maps f,g:(M,p)→M′, there exists an integer k=kp such that if f and g have the same k-jets at p, and do not send M into EM′, then necessarily f=g. Furthermore, the map p↦kp may be chosen to be bounded on compact subsets of M. As a consequence, we derive the finite jet determination property for pairs of germs of CR maps from minimal real-analytic CR submanifolds in CN into Nash subsets in CN′ of D'Angelo finite type, for arbitrary N,N′≥2.

Double step heating synthesis of MIL-101(Cr) composites for water harvesting applications

Energy storage is a valuable mechanism for managing fluctuations in energy demand by allowing the distribution of energy production needed during peak demand over a more extended period. Thermal energy storage is among the most valuable categories of energy storage. The optimal sorbent material must satisfy thermal, physical, chemical, and environmental requirements for designing sorption heat storage systems. MIL-101(Cr) is a Metal-Organic Framework (MOF) with remarkable properties such as a high surface area, hydrothermal stability and good water sorbent capacity that make it a potential candidate as a working sorbent. Their physicochemical properties can be modulated by functionalizing the organic linker or by modifying its structure, improving the interaction with the target sorbate, mainly environmentally friendly and non-toxic sorbates such as water. However, good water sorption capacity is not the only characteristic to be considered when selecting the sorbent material for sorption heat storage systems. Here, we propose an alternative method to obtain a sorbent material with the objective of reducing the reaction time and achieving a good balance between the physical properties of the material, such as the density and the thermal conductivity, without substantially decreasing its water uptake capacity. For this aim, we show that structural modification of MIL-101(Cr) induced by CaCl2 and graphene oxide (GO) can improve the water sorption capacity and optical properties compared to the pristine sorbent. The MIL-101(Cr)/GO/CaCl2 composite had a higher water uptake capacity of 1.08 gwater g−1composite-1, even compared with MIL-101(Cr)/CaCl2 composite with a water uptake capacity of 0.68 gwater g−1composite. Additionally, MIL-101(Cr)/GO and MIL-101(Cr)/GO/CaCl2 sorbents tend to absorb light toward the near-infrared region and increase the temperature around twice as much compared to sorbents without GO when illuminated with a flash lamp. These results demonstrate that GO added at the MIL-101(Cr) structure allows a better distribution of CaCl2 in the MIL-101(Cr) pores and it may be advantageous in the water desorption process making use of renewable infrared radiation sources such as solar energy.

Paley–Wiener–Schwartz theorems on quadratic CR manifolds

Paley–Wiener–Schwartz theorems on quadratic CR manifolds

The total Cartan curvature of Reinhardt surfaces

The Cartan umbilical tensor is a crucial biholomorphic invariant for three-dimensional strictly pseudoconvex CR manifolds. When considering compact manifolds, its L1-norm, referred to as the total Cartan curvature, is an important global numerical invariant. In this study, we seek to calculate the total Cartan curvature of T3, the three-torus, which features a specific T2-invariant CR structure, incorporating Reinhardt boundaries with closed logarithmic generating curves as a special instance. Our findings present an unexpected corollary indicating that, for many of these boundaries, the total Cartan curvature is equal to 8π2 times the Burns–Epstein invariant.

Carleson and sampling measures on Bernstein spaces on Siegel CR manifolds

AbstractIn this paper, we introduce and study Carleson and sampling measures on Bernstein spaces on a class of quadratic CR manifold called Siegel CR manifolds. These are spaces of entire functions of exponential type whose restrictions to the given Siegel CR manifold are ‐integrable with respect to a natural measure. For these spaces, we prove necessary and sufficient conditions for a Radon measure to be a Carleson or a sampling measure. We also provide sufficient conditions for sampling sequences.

Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on CR manifolds

We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented pseudohermitian structures \mathcal{P}_+ . We show that the functionals are continuous with respect to a natural topology on \mathcal{P}_+ . Using an adaptation of the standard Kato–Rellich perturbation theory, we prove that the functionals are (one-sided) differentiable along 1-parameter analytic deformations. We use this differentiability to define the notion of critical pseudohermitian structures, in a generalized sense, for them. We give a necessary (also sufficient in some situations) condition for a pseudohermitian structure to be critical. Finally, we present explicit examples of critical pseudohermitian structures on both homogeneous and non-homogeneous CR manifolds.

Homogeneous CR submanifolds of complex hyperbolic spaces

Homogeneous CR submanifolds of complex hyperbolic spaces

Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds

Deformations and embeddings of three-dimensional strictly pseudoconvex CR manifolds

Levi-flat CR structures on compact Lie groups

Pittie (Proc Indian Acad Sci Math Sci 98:117-152, 1988) proved that the Dolbeault cohomology of all left-invariant complex structures on compact Lie groups can be computed by looking at the Dolbeault cohomology induced on a conveniently chosen maximal torus. We generalized Pittie’s result to left-invariant Levi-flat CR structures of maximal rank on compact Lie groups. The main tools we used was a version of the Leray–Hirsch theorem for CR principal bundles and the algebraic classification of left-invariant CR structures of maximal rank on compact Lie groups (Charbonnel and Khalgui in J Lie Theory 14:165-198, 2004) .

First principles investigation of 3d transition metal doped SnO monolayer based diluted magnetic semiconductors

The two-dimensional (2D) SnO monolayer attracted capacious recognition on account of its incomparable applications in nanoelectronics. Diluted magnetic semiconductors (DMS) are realized via doping of transition metals (TM) in semiconductors for usage in spintronic devices. This work is carried out using first-principles method to substitute 3d TM impurity atoms (Sc, Ti, V, Cr, Co, Ni, Cu and Zn) in place of Sn cation sites in the SnO monolayers. The analysis of modification resulted after the doping process is carried out to investigate the changes in the structural, electronic and magnetic properties of the materials. The calculated formation energy is utilized to examine the structural stability of the doped materials. The material doped with Sc, Ti, V, Cr, Co and Cu showed magnetic behavior while no magnetization is detected in case of doping with Ni and Zn. The results are discussed in the light of calculated crystal field splitting and exchange interactions between the two impurity-doped atoms. The magnetic ground state and Curie temperature of the DMS are also estimated. The Sc, V and Co doping has a ferromagnetic ground state while Ti, Cr and Cu doped structures got the AFM ground state. The mixing of TM-3d orbitals with the host s and p states is considered responsible for magnetism and the magnetic moment. The outcomes of the study point out that the TM doped SnO monolayer based DMS may be helpful to realize the spin-TFT devices in the near future.

On finitely nondegenerate closed homogeneous CR manifolds

A complex flag manifold $$\textsf {F}{=}{{\textbf {G}}}/{{\textbf {Q}}}$$ decomposes into finitely many real orbits under the action of a real form $${{\textbf {G}}}^\upsigma $$ of $${{\textbf {G}}}$$ . Their embedding into $$\textsf {F}$$ defines on them CR manifold structures. We characterize and list all the closed real orbits which are finitely nondegenerate.

Generalizations of the Q-prime curvature via renormalized characteristic forms

The Q-prime curvature is a local pseudo-Einstein invariant on CR manifolds defined by Case and Yang, and Hirachi. Its integral, the total Q-prime curvature, gives a non-trivial global CR invariant. On the other hand, Marugame has constructed a family of global CR invariants via renormalized characteristic forms, which contains the total Q-prime curvature. In this paper, we introduce a generalization of the Q-prime curvature for each renormalized characteristic form, and show that its integral coincides with Marugame's CR invariant. We also study generalizations of the critical CR GJMS operator and the P-prime operator, which are related to the transformation laws of our new curvatures under conformal change.

C-R Immersions and Sub-Riemannian Geometry

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form θ, we consider natural ϵ-contractions, i.e., contractions gϵM of the Levi form Gθ, such that the norm of the Reeb vector field T of (M, θ) is of order O(ϵ−1). We study isopseudohermitian (i.e., f∗Θ=θ) Cauchy–Riemann immersions f:M→(A,Θ) between strictly pseudoconvex CR manifolds M and A, where Θ is a contact form on A. For every contraction gϵA of the Levi form GΘ, we write the embedding equations for the immersion f:M→A,gϵA. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as ϵ→0+. For every isopseudohermitian immersion f:M→S2N+1 into a sphere S2N+1⊂CN+1, we show that Webster’s pseudohermitian scalar curvature R of (M, θ) satisfies the inequality R≤2n(f∗gΘ)(T,T)+n+1+12{∥H(f)∥gΘf2+∥traceGθΠH(M)∇⊤−∇∥f∗gΘ2} with equality if and only if B(f)=0 and ∇⊤=∇ on H(M)⊗H(M). This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms.

On the Sobolev quotient of three-dimensional CR manifolds

We exhibit examples of compact three-dimensional CR manifolds of positive Webster class, Rossi spheres , for which the pseudo-hermitian mass, as defined by Cheng–Malchiodi–Yang (2017), is negative, and for which the infimum of the CR-Sobolev quotient is not attained. To our knowledge, this is the first geometric context on smooth closed manifolds where this phenomenon arises, in striking contrast to the Riemannian case.

Improved higher-order Sobolev inequalities on CR sphere

We improve higher-order CR Sobolev inequalities on S2n+1 under the vanishing of higher order moments of the volume element. As an application, we give a new and direct proof of the classification of minimizers of the CR invariant higher-order Sobolev inequalities. In the same spirit, we prove almost sharp Sobolev inequalities for GJMS operators to general CR manifolds, and obtain the existence of minimizers in C2k(N) of higher-order CR Yamabe-type problems when Yk(N)<Yk(Hn).

Nonlinearizable CR Automorphisms for Polynomial Models in $$\mathbb C^N$$

The Lie algebra of infinitesimal CR automorphisms is a fundamental local invariant of a CR manifold. Motivated by the Poincaré local equivalence problem, we analyze its positively graded components, containing nonlinearizable holomorphic vector fields. The results provide a complete description of invariant weighted homogeneous polynomial models in $$\mathbb C^N$$ , which admit symmetries of degree higher than two. For homogeneous polynomial models, symmetries with quadratic coefficients are also classified completely. As a consequence, this provides an optimal 1-jet determination result in the general case. Further we prove that such automorphisms arise from one common source, by pulling back via a holomorphic mapping a suitable symmetry of a hyperquadric in some (typically high dimensional) complex space.

$\mathcal{I}^{\prime}$-curvatures in higher dimensions and the Hirachi conjecture

We construct higher-dimensional analogues of the $\mathcal{I}^{\prime}$-curvature of Case and Gover in all CR dimensions $n \geq 2$. Our $\mathcal{I}^{\prime}$-curvatures all transform by a first-order linear differential operator under a change of contact form and their total integrals are independent of the choice of pseudo-Einstein contact form on a closed CR manifold. We exhibit examples where these total integrals depend on the choice of general contact form, and thereby produce counterexamples to the Hirachi conjecture in all CR dimensions $n \geq 2$.