Spinorial Characterization of Surfaces into 3-dimensional homogeneous Manifolds

We give a spinorial characterization of isometrically immersed surfaces into 3-dimensional homogeneous manifolds with 4-dimensional isometry group in terms of the existence of a particular spinor, called generalized Killing spinor. This generalizes results by T. Friedrich for $\R^3$ and B. Morel for \Ss^3 and \HH^3. The main argument is the interpretation of the energy-momentum tensor of a genralized Killing spinor as the second fondamental form up to a tensor depending on the structure of the ambient spaceComment: 35 page

The Hijazi inequality on manifolds with boundary

In this paper, we prove the Hijazi inequality on compact Riemannian spin manifolds under two boundary conditions: the condition associated with a chirality operator and the Riemannian version of the \MIT bag condition. We then show that the limiting-case is characterized as being a half-sphere for the first condition whereas the equality cannot be achieved for the second.Comment: 14 page

Pricing Spread Options using Matched Asymptotic Expansions

This document deals with approximating spread options prices using Matched Asymptotic Expansions techniques on the correlation. More precisely, it deals with spreads options on assets that are highly correlated (ρ ∼ 1), which is most commonly observed in Oil Markets (Crude Oil vs. Gasoline for example). We will first start by applying this methodology to exchange options before generalizing our results to spread options. Then we are going to describe an alternative approach of pricing spread options by approximating the bivariate normal distribution. Finally, we will see how we can apply our methodology to the case where we have more than two assets

Prefabricated foldable lunar base modular systems for habitats, offices, and laboratories

The first habitat and work station on the lunar surface undoubtedly has to be prefabricated, self-erecting, and self-contained. The building structure should be folded and compacted to the minimum size and made of materials of minimum weight. It must also be designed to provide maximum possible habitable and usable space on the Moon. For this purpose the concept of multistory, foldable structures was further developed. The idea is to contain foldable structural units in a cylinder or in a capsule adapted for launching. Upon landing on the lunar surface, the cylinder of the first proposal in this paper will open in two hinge-connected halves while the capsule of the second proposal will expand horizontally and vertically in all directions. In both proposals, the foldable structural units will self-erect providing a multistory building with several room enclosures. The solar radiation protection is maintained through regolith-filled pneumatic structures as in the first proposal, or two regolith-filled expandable capsule shells as in the second one, which provide the shielding while being supported by the erected internal skeletal structure

Eigenvalue estimates for the Dirac-Schr\"odinger operators

We give new estimates for the eigenvalues of the hypersurface Dirac operator in terms of the intrinsic energy-momentum tensor, the mean curvature and the scalar curvature. We also discuss their limiting cases as well as the limiting cases of the estimates obtained by X. Zhang and O. Hijazi in [13] and [10]. We compare these limiting cases with those corresponding to the Friedrich and Hijazi inequalities. We conclude by comparing these results to intrinsic estimates for the Dirac-Schr\"odinger operator D_f = D - f/2.Comment: 22 pages, LaTeX, to appear in Journal of Geometry and Physic

A holographic principle for the existence of parallel spinor fields and an inequality of Shi-Tam type

Suppose that $\Sigma=\partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $( M,\langle\;,\;\rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $\Sigma$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric $\langle\;,\;\rangle_H=H^2\langle\;,\;\rangle$ is at least $n/2$ and equality holds if and only if there exists a parallel spinor field on $M$. As a consequence, if $\Sigma$ admits an isometric and isospin immersion $\phi$ with mean curvature $H_0$ as a hypersurface into another spin Riemannian manifold $M_0$ admitting a parallel spinor field, then \begin{equation} \label{HoloIneq} \int_\Sigma H\,d\Sigma\le \int_\Sigma \frac{H^2_0}{H}\, d\Sigma \end{equation} and equality holds if and only if both immersions have the same shape operator. In this case, $\Sigma$ has to be also connected. In the special case where $M_0=\R^{n+1}$, equality in (\ref{HoloIneq}) implies that $M$ is an Euclidean domain and $\phi$ is congruent to the embedding of $\Sigma$ in $M$ as its boundary. We also prove that Inequality (\ref{HoloIneq}) implies the Positive Mass Theorem (PMT). Note that, using the PMT and the additional assumption that $\phi$ is a strictly convex embedding into the Euclidean space, Shi and Tam \cite{ST1} proved the integral inequality \begin{equation}\label{shi-tam-Ineq} \int_\Sigma H\,d\Sigma\le \int_\Sigma H_0\, d\Sigma, \end{equation} which is stronger than (\ref{HoloIneq}) .Comment: arXiv admin note: text overlap with arXiv:1502.0408