Christian Bär

Real Killing spinors and holonomy

We give a description of all complete simply connected Riemannian manifolds carrying real Killing spinors. Furthermore, we present a construction method for manifolds with the exceptional holonomy groupsG2 and Spin(7).

Wave equations on Lorentzian manifolds and quantization

This book provides a detailed introduction to linear wave equations on Lorentzian manifolds (for vector-bundle valued fields). After a collection of preliminary material in the first chapter one finds in the second chapter the construction of local fundamental solutions together with their Hadamard expansion. The third chapter establishes the existence and uniqueness of global fundamental solutions on globally hyperbolic spacetimes and discusses Greens operators and well-posedness of the Cauchy problem. The last chapter is devoted to field quantization in the sense of algebraic quantum field theory. The necessary basics on C*-algebras and CCR-representations are developed in full detail. The text provides a self-contained introduction to these topics addressed to graduate students in mathematics and physics. At the same time it is intended as a reference for researchers in global analysis, general relativity, and quantum field theory.

Extrinsic Bounds for Eigenvalues of the Dirac Operator

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.

Metrics with harmonic spinors

We show that every closed spin manifold of dimensionn ≡ 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheresSn,n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sumM1#M2 with certain metrics is close to the union of the spectra ofM1 and ofM2.

Lower eigenvalue estimates for Dirac operators

We derive upper eigenvalue estimates for generalized Dirac operators on closed Riemannian manifolds. In the case of the classical Dirac operator the estimates on the first eigenvalues are sharp for spheres of constant curvature.

Generalized cylinders in semi-Riemannian and spin geometry

Abstract.We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.

On Nodal Sets for Dirac and Laplace Operators

Abstract:We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator on a Riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension. We also see that the nodal set of a Δ-harmonic differential form on a closed manifold has codimension 2 at least; a fact which is not true if the manifold is not closed. Examples show that all bounds are optimal.

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ± ∞ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities.Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.

The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces

The paper “The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces”, Arch. Math. 59, 65–79 (1992), contains some misprints which I would like to correct in this erratum. One important point: In Theorem 5 on p. 79 (and also on p. 78, l. 6) replace [−(m + 1)/N ] < i by −[(m − 2)/N ] ≤ i. Similarly, replace [(1 − m − N ′)/N ] < i in Theorem 5 by −[(m− 2 + N ′)/N ] ≤ i. Some fairly obvious misprints: • p. 66, l. 12: Replace [g, b̄0 ·Θ(Λ)] by [g,Θ(Λ)]. • p. 68, l. 5: Replace (dg0 · X̄)so([g0, b̄0]) = d dt [g0e tX , b̄0(Θ · Λ(t))]|t=0 by (dg0 · X̄)so([g0, 1so]) = d dt [g0e tX ,Θ · Λ(t)]|t=0. • p. 68, l. 20: Replace ∇ dt [e tX , b̄0 · (Θ · Λ(t))]|t=0 by ∇ dt [e tX ,Θ · Λ(t)]|t=0. • p. 69, l. 14: Add |t=0. • p. 71, l. 6: Replace Aπγ(g)v by Aπγ(g)v. • p. 72, l. 2: Replace E3 = 1 T ( i 0 0 −i )

Zero sets of solutions to semilinear elliptic systems of first order

Abstract.Consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n−2)-dimensional submanifolds. Hence it is countably (n−2)-rectifiable and its Hausdorff dimension is at most n−2. Moreover, it has locally finite (n−2)-dimensional Hausdorff measure. We show by example that every real number between 0 and n−2 actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic equations of second order.