On the first eigenvalue of the Dirichlet-to-Neumann operator on forms

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.Comment: 26 page

A Reilly formula and eigenvalue estimates for differential forms

We derive a Reilly-type formula for differential p-forms on a compact manifold with boundary and apply it to give a sharp lower bound of the spectrum of the Hodge Laplacian acting on differential forms of an embedded hypersurface of a Riemannian manifold. The equality case of our inequality gives rise to a number of rigidity results, when the geometry of the boundary has special properties and the domain is non-negatively curved. Finally we also obtain, as a by-product of our calculations, an upper bound of the first eigenvalue of the Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.Comment: 22 page

Branson's Q-curvature in Riemannian and Spin Geometry

On a closed n-dimensional manifold, n ≥ 5, we compare the three basic conformally covariant operators: the Paneitz-Branson, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's Q-curvature. Equality cases are also characterized

On a spin conformal invariant on manifolds with boundary

On a n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric, a spin structure and a chirality operator, we study some properties of a spin conformal invariant defined from the first eigenvalue of the Dirac operator under the chiral bag boundary condition. More precisely, we show that we can derive a spinorial analogue of Aubin's inequality.Comment: 26 page

The Dirac operator on untrapped surfaces

We establish a sharp extrinsic lower bound for the first eigenvalue of the Dirac operator of an untrapped surface in initial data sets without apparent horizon in terms of the norm of its mean curvature vector. The equality case leads to rigidity results for the constraint equations with spherical boundary as well as uniqueness results for constant mean curvature surfaces in Minkowski space.Comment: 16 page

Optimal eigenvalues estimate for the Dirac operator on domains with boundary

We give a lower bound for the eigenvalues of the Dirac operator on a compact domain of a Riemannian spin manifold under the \MIT bag boundary condition. The limiting case is characterized by the existence of an imaginary Killing spinor.Comment: 10 page

Optimal eigenvalue estimate for the Dirac-Witten operator on bounded domains with smooth boundary

Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for the usual Dirac operator. The limiting cases are also investigated.Comment: 2007, 18 pages, submitted 02 June 200

Rigidity of compact Riemannian spin Manifolds with Boundary

In this article, we prove new rigidity results for compact Riemannian spin manifolds with boundary whose scalar curvature is bounded from below by a non-positive constant. In particular, we obtain generalizations of a result of Hang-Wang \cite{hangwang1} based on a conjecture of Schroeder and Strake \cite{schroeder}.Comment: English version of "G\'eom\'etrie spinorielle extrins\`eque et rigidit\'es", Corollary 6 in Section 3 added, to appear in Letters Math. Phy

Elastic and vibrational properties of alpha and beta-PbO

The structure, electronic and dynamic properties of the two layered alpha (litharge) and beta (massicot) phases of PbO have been studied by density functional methods. The role of London dispersion interactions as leading component of the total interaction energy between layers has been addressed by using the Grimme's approach, in which new parameters for Pb and O atoms have been developed. Both gradient corrected and hybrid functionals have been adopted using Gaussian-type basis sets of polarized triple zeta quality for O atoms and small core pseudo-potential for the Pb atoms. Basis set superposition error (BSSE) has been accounted for by the Boys-Bernardi correction to compute the interlayer separation. Cross check with calculations adopting plane waves that are BSSE free have also been performed for both structures and vibrational frequencies. With the new set of proposed Grimme's type parameters structures and dynamical parameters for both PbO phases are in good agreement with experimental data.Comment: 8 pages, 5 figure