Extrinsic Killing spinors

 Under intrinsic and extrinsic curvature assumptions on a Riemannian spin manifold and its boundary, we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic Killing spinors of non-negative Killing constant. As a corollary, we prove that a complete Ricci-flat spin manifold with mean-convex boundary isometric to a round sphere, is necessarily a flat disc. Received: 2 February 2002; in final form: 1 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (1991): 53C27, 53C40, 53C80, 58G25 The authors would like to thank Lars Andersson for helpful discussions and for bringing to our knowledge the information regarding Remark 4. We are also grateful to the referee for pointing out that Corollary 5 and Corollary 6 are only valid when the boundary is at least 2-dimensional. Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001-2967     

Brightly phosphorescent trinuclear copper(I) complexes of pyrazolates: substituent effects on the supramolecular structure and photophysics

Synthetic details, solid-state structures, and photophysical properties of a group of trimeric copper(I) complexes containing pyrazolate ligands are described. The reaction of copper(I) oxide and the fluorinated pyrazoles [3-(CF(3))Pz]H, [3-(CF(3)),5-(Me)Pz]H, and [3-(CF(3)),5-(Ph)Pz]H leads to the corresponding trinuclear copper(I) pyrazolates, {[3-(CF(3))Pz]Cu}(3), {[3-(CF(3)),5-(Me)Pz]Cu}(3), and {[3-(CF(3)),5-(Ph)Pz]Cu}(3), respectively, in high yield. The {[3,5-(i-Pr)(2)Pz]Cu}(3) compound was obtained by a reaction between [Cu(CH(3)CN)(4)][BF(4)], [3,5-(i-Pr)(2)Pz]H, and NEt(3). These compounds as well as {[3,5-(Me)(2)Pz]Cu}(3) and {[3,5-(CF(3))(2)Pz]Cu}(3) adopt trimeric structures with nine-membered Cu(3)N(6) metallacycles. There are varying degrees and types of intertrimer Cu...Cu interactions. These contacts give rise to zigzag chains in the fluorinated complexes, {[3-(CF(3))Pz]Cu}(3), {[3-(CF(3)),5-(Me)Pz]Cu}(3), {[3-(CF(3)),5-(Ph)Pz]Cu}(3), and {[3,5-(CF(3))(2)Pz]Cu}(3), whereas the nonfluorinated complexes, {[3,5-(Me)(2)Pz]Cu}(3) and {[3,5-(i-Pr)(2)Pz]Cu}(3) form dimers of trimers. Out of all the compounds examined in this study, {[3-(CF(3)),5-(Ph)Pz]Cu}(3) has the longest (3.848 Angstroms) and {[3,5-(Me)(2)Pz]Cu}(3) has the shortest (2.946 Angstroms) next-neighbor intertrimer Cu...Cu distance. The Cu...Cu separations within the trimer units do not vary significantly (typically 3.20-3.26 Angstroms). All of these trinuclear copper(I) pyrazolates show bright luminescence upon exposure to UV radiation. The luminescence bands are hugely red-shifted from the corresponding lowest-energy excitations, rather broad, and unstructured even at low temperatures, suggesting metal-centered emissions owing to intertrimer Cu...Cu interactions that are strengthened in the phosphorescent state. The {[3-(CF(3)),5-(Ph)Pz]Cu}(3) compound exhibits an additional highly structured phosphorescence with a vibronic structure corresponding to the pyrazolyl (Pz) ring. The luminescence properties of solids and solutions of the trimeric compounds in this study show fascinating trends with dramatic sensitivities to temperature, solvent, concentration, and excitation wavelengths.     

Development of new high‐performance visible light photoinitiators based on carbazole scaffold and their applications in 3d printing and photocomposite synthesis

In this article, new compounds based on the carbazole scaffold (DMs = DM1 and DM2, constituted by a carbazole unit connected on positions 3 and 6 to a two 4,4′‐dimethoxydiphenylamine groups and differing by the substituent present on the nitrogen heteroatom of the carbazole core) were synthesized and proposed as high‐performance visible light photoinitiators/photosensitizers for both the free‐radical polymerization of methacrylates and the cationic polymerization of epoxides upon visible light exposure using LED@405 nm. Remarkably, DM2 leads to higher final conversions than DM1. In order to study the photophysical and photochemical properties of the carbazole derivatives, different parameters were taken into account such as the light absorption, the steady‐state photolysis, and the fluorescence spectroscopy. Using different techniques such as fluorescence quenching, redox behavior, and cyclic voltammetry, we are able to discuss the photosensitization/photoinitiation reactions providing a full coherent picture of the involved chemical mechanisms. The photosensitization of the carbazole derivatives occurred predominantly via singlet excited states at the rate of the diffusion limit. Upon exposure to laser diode at 405 nm, DMs show high performance in initiating systems for 3D resins. Remarkably, DM2 can also be used in photocomposite synthesis using light‐emitting diode conveyor. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019, 57, 2081–2092     

Photoconductivity,Antioxidant, and Antimicrobial Activities of Some Acenaphthenequinone Derivatives

Russian Journal of General Chemistry - Photoconductive acenaphthenequinone derivatives have been synthesized by a one-pot process of acenaphthenequinone with different aromatic hydrazides...     

Dirac Operators on Hypersurfaces of Manifolds with Negative Scalar Curvature

We give a sharp extrinsic lower bound for the first eigenvaluesof the intrinsic Dirac operator of certain hypersurfaces boundinga compact domain in a spin manifold of negative scalar curvature.Limiting-cases are characterized by the existence, on the domain,of imaginary Killing spinors. Some geometrical applications, as anAlexandrov type theorem, are given.     

Eigenvalues of the Kählerian Dirac Operator

We get estimates on the eigenvalues of the Kählerian Dirac operator in terms of the eigenvalues of the scalar Laplace–Beltrami operator. In odd complex dimension, these estimates are sharp, in the sense that, for the first eigenvalue, they reduce to Kirchberg's inequality.     

Trapping Nanostructures on Surfaces through Weak Interactions

The assembly of imidazole‐functionalized phenanthroline‐strapped zinc porphyrins (ZnPorphen) with alkyl or polyethylene glycol (PEG) side chains was studied in solution and by AFM after casting on highly oriented pyrolytic graphite (HOPG) or mica. The nature of the solvent and its evaporation time influenced the morphology of the objects observed. On HOPG, short rods of about 100 nm were observed after fast evaporation of solutions of the alkyl derivatives in CHCl₃, THF, or pyridine, whereas islands of aligned rows of longer wires were obtained from methylcyclohexane (MCH). Slow evaporation of MCH led to a three‐dimensional assembly. The PEG porphyrin assembled into short wires on HOPG or fibers on mica after slow evaporation of solutions in THF. This study shows the role of surface–molecule interactions in the interfacial assembly of ZnPorphen derivatives and contributes to understanding the parameters that control their noncovalent assembly into molecular wires on a surface.     

Capture of the complex [Ni(dto)2]2- (dto2- = dithiooxalato ligand) in a Mo12 ring: synthesis, characterizations, and application toward the reduction of protons

The encapsulation of the complex [Ni(dto)(2)](2-) within an oxothiododecamolybdic cyclic cluster has been investigated. The resulting molybdenum ring, [Mo(12)O(12)S(12)(OH)(12)(Ni(dto)(2))](2-), corresponds to the first example of the {Mo(2)O(2)S(2)}-based assembly arranged around a 3d transition-metal complex. It was unambiguously characterized in the solid state and in solution by FT-IR spectroscopy, single-crystal X-ray diffraction, NMR, UV-visible spectroscopy, and electrospray ionization-high-resolution mass spectrometry (ESI-HRMS). The latter technique revealed to be a powerful tool for the characterization of templated molybdenum ring systems in solution and gave excellent results in high resolution. The electronic spectrum of [Mo(12)O(12)S(12)(OH)(12)(Ni(dto)(2))](2-) evidenced a strong red shift of the LMCT bands attributed to the complex [Ni(dto)(2)](2-), suggesting significant variations of the electronic properties upon its encapsulation within the Mo(12) ring. These differences were demonstrated by electrochemical studies in CH(3)CN, which also revealed, for both compounds [Ni(dto)(2)](2-) and [Mo(12)O(12)S(12)(OH)(12)(Ni(dto)(2))](2-), electrocatalytic properties for the reduction of protons. These results evidence the ability of dithioxalato complexes to act as electrocatalysts for the hydrogen evolution reaction (HER) and confirm such a property for oxothiomolybdenum rings.     

On/off luminescence vapochromic selective sensing of benzene and its methylated derivatives by a trinuclear silver(I) pyrazolate sensor

{[3,5-(CF(3))(2)Pz]Ag}(3) (1) films exhibit selective/reversible sensing of small-organic-molecule (SAM) vapors, which readily switch-on bright-green (benzene or toluene) or bright-blue (mesitylene) luminescence that switches-off upon vapor removal. Vapors of electron-deficient SAMs or non-aromatic solvents did not attain luminescence switching and were not adsorbed.