BRS-Transformations in a Finite Dimensional Setting, proceedings of the 16th winter school „geometry and physics“, 1997

In order to get a mathematical understanding of the BRS-transformation and the Slavnov-Taylor identities, we will treat them in a finite dimensional setting. We will show that in this setting the BRS-transformation is a vector field on a certain supermanifold. The connection to the BRS-complex will be established. Finally we will treat the generating functional and the Slavnov-Taylor identity in this setting.

Lower bounds for eigenvalues of the Dirac operator on surfaces of rotation, Journal of Geometry and Physics 31 (1999)

In this paper we will prove new lower bounds or the first eigenvalue of the Dirac operator on 2-dimensional Riemannian manifolds iffeomorphic to S² with an isometric S¹-action. We how examples, where this new bound improves the known lower bounds and coincides in the limit with the known upper bounds.

Lower bounds for eigenvalues of the Dirac operator on n-spheres with SO(n)-symmetry, Journal of Geometry and Physics 32 (2000)

In this paper we derive estimates for the eigenvalues of the Dirac operator and their multiplicity on manifolds diffeomorphic to Sⁿ with an isometric SO(n)-action. Especially we prove a new lower bound for the first eigenvalue and show an example, where this new bound coincides in the limit with the known upper bounds.

Eigenvalues of the Dirac operator on fibrations over S¹, Annals of Global Analysis and Geometry 18, 2000

We consider the Dirac operator on fibrations over S¹ which have up to holonomy a warped product metric. We give lower bounds for the eigenvalues on M and if the Dirac operator on the typical fibre F has a kernel, we calculate the corresponding part of the spectrum on M explicitly. Moreover we discuss the dependence of the spectrum of the holonomy and obtain bounds for the multiplicity of the eigenvalues.

A new method for eigenvalue estimates for Dirac operators on certain manifolds with SO(k)-symmetry, Differential Geometry and Applications 19 (2003), no 1

We prove estimates for the first nonnegative eigenvalue of the Dirac operator on certain manifolds with SO(k+1)-symmetry in terms of geometric properties of the manifold. For the  proof we employ an abstract technique which is new in this context and may apply to other cases of manifolds as well.

Asymptotic estimates for Dirac and Laplace eigenvalues on warped products over S¹, erscheint in manuscripta mathematica

In this paper we show that the space of spinors over a warped product over S¹ has a certain splitting $\Gamma \Sigma = \oplus W(k,n)$ in spaces of spinors of weight k and winding number n which is respected by the Dirac operator. The same holds for the space of functions and the Laplace operator. We give eigenvalue estimates for the eigenvalues $\lambda_{k,n}$ of the Dirac operator with eigenspinors of weight k and winding number n and eigenvalues $\mu_{m,n}$ of the Laplace operator with eigenfunctions of weight m and winding number n. In particular, we show that $\mu_{k^2,n} \geq \lambda^2_{k,n}$ holds for large n on $S^1 \times_f T^l$ where $T^l$ is a flat torus with the trivial spin structure.

Variational principles for Dirac operators, erscheint in Journ. Comp. and Appl. Math.

In this paper we establish variational principles, eigenvalue estimates and asymptotic formulae for eigenvalues of three different classes of unbounded block operator matrices. The results allow to characterise eigenvalues that are not necessarily located at the boundary of the spectrum. Applications to an example from magnetohydrodynamics and to Dirac operators on certain manifolds are given.

Curvature Estimates for Asymptotically Flat Hypersurfaces of Lorentzian Manifolds, erscheint in Can. Journ. Math.

We consider an asymptotically flat Lorentzian manifold of dimension (1,3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when the ADM energy tends to zero.

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