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     

Subordination des résolvantes

Résumé Soit (V ₎₀ une résolvante définie sur un espace mesurable telle que le noyau initial est borné; on trouve une condition nécéssaire et suffisante pour qu'un noyau borné U possède une résolvante (U ₎₀ telle que U V pour tout 0. On donne plusieurs applications de ce résultat.     

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.     

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.     

Tunable THz metamaterials based on an array of paraelectric SrTiO<Subscript>3</Subscript> rods

This work presents theoretical and experimental investigations of a tunable metamaterial which exhibits negative permeability in the THz frequency range. The tunability is obtained by temperature changes, and the sample consists of an array of high-permittivity SrTiO₃ (STO) rods micromachined by a femtosecond laser. Structures exhibiting a negative permeability on multiple frequency bands are also investigated and a proper choice of the dimensions of the pattern allows us to achieve a substantial broadening of the frequency band with negative μ.     

3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope

Microscopic interferometry is a powerful technique for the static and dynamic characterization of micromechanical devices. In this paper we emphasize its capabilities for 3D vibration mode shapes profiling of Al cantilever microbeams and Cr micromachined membranes. It is demonstrated that time-resolved measurements up to 800 kHz can be performed with a lateral resolution in the micrometer range and a vibration amplitude detection limit of 3–5 nm. In addition, with reduced image sizes (256×256), quasi-real time (150–500 ms) visualization of the vibration mode 3D profiles becomes possible. These performances were obtained by using stroboscopic illumination with an array of superluminescent LED and an optimized automatic fast Fourier transform phase demodulation of the interferograms. The results are compared with theoretical shapes of the vibration modes and with point measurements of the vibration spectra.     

Chelation-mediated palladium(II)-catalyzed domino Heck-Mizoroki/Suzuki-Miyaura reactions using arylboronic acids: increasing scope and mechanistic understanding

A palladium(II)-catalyzed Heck-Mizoroki/Suzuki-Miyaura domino reaction involving metal coordinating dimethylaminoethyl vinyl ethers and a number of electron-rich and electron-deficient arylboronic acids has been developed. Through variation of the temperature and the concentration of the p-benzoquinone (p-Bq) ligand/reoxidant, conditions for the robust and convenient one-pot generation of diarylated-saturated ethers were identified. With the aid of coordination of the dimethylamino group to the arylpalladium intermediate, the otherwise predominant formation of the β-arylated olefin could be reversed. A reaction route involving a chelation-controlled carbopalladation, providing a p-Bq stabilized six-membered palladacycle, followed by transmetalation and reductive elimination is suggested to explain the selective formation of saturated diarylated ether products.     

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.