TY - JOUR A1 - Bär, Christian A1 - Wafo, Roger Tagne T1 - Initial value problems for wave equations on manifolds JF - Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics N2 - We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hormander. KW - Wave equation KW - Globally hyperbolic Lorentz manifold KW - Cauchy problem KW - Goursat problem KW - Finite energy sections Y1 - 2015 U6 - https://doi.org/10.1007/s11040-015-9176-7 SN - 1385-0172 SN - 1572-9656 VL - 18 IS - 1 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Bär, Christian A1 - Strohmaier, Alexander T1 - An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary JF - American Journal of Mathematics N2 - We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed. Y1 - 2019 U6 - https://doi.org/10.1353/ajm.2019.0037 SN - 0002-9327 SN - 1080-6377 VL - 141 IS - 5 SP - 1421 EP - 1455 PB - Johns Hopkins Univ. Press CY - Baltimore ER - TY - JOUR A1 - Bär, Christian A1 - Strohmaier, Alexander T1 - A Rigorous Geometric Derivation of the Chiral Anomaly in Curved Backgrounds JF - Communications in mathematical physics N2 - We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived directly in Lorentzian signature and in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the.-invariant of the Cauchy hypersurfaces. Y1 - 2016 U6 - https://doi.org/10.1007/s00220-016-2664-1 SN - 0010-3616 SN - 1432-0916 VL - 347 SP - 703 EP - 721 PB - Springer CY - New York ER - TY - JOUR A1 - Bär, Christian A1 - Schopka, Sven T1 - The dirac determinant of spherical space forms N2 - The zeta-regularized determinants of the Dirac operator and of its square are computed on spherical space forms. On S^2 the determinant of Dirac operators twisted by a complex line bundle is also calculated. Y1 - 2003 UR - http://users.math.uni-potsdam.de/~baer/preprints/determinante.html ER - TY - INPR A1 - Bär, Christian A1 - Pfäffle, Frank T1 - Wiener measures on Riemannian manifolds and the Feynman-Kac formula N2 - This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)17 KW - Wiener measure KW - conditional Wiener measure KW - Brownian motion KW - Brownian bridge KW - Riemannian manifold Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-59998 ER - TY - JOUR A1 - Bär, Christian A1 - Pfaeffle, Frank T1 - Asymptotic heat kernel expansion in the semi-classical limit N2 - Let H-h = h(2)L + V, where L is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and V is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of H-h as h SE arrow 0. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive h by the classical partition function. Y1 - 2010 UR - http://www.springerlink.com/content/100467 U6 - https://doi.org/10.1007/s00220-009-0973-3 SN - 0010-3616 ER - TY - JOUR A1 - Bär, Christian A1 - Moroianu, Sergiu T1 - Heat kernel asymptotics for roots of generalized laplacians N2 - We describe the heat kernel asymptotics for roots of a Laplace type operator on a closed manifold. A previously known relation between the Wodzicki residue and heat trace asymptotics is shown to hold pointwise for the corresponding densities. Y1 - 2003 UR - http://users.math.uni-potsdam.de/~baer/preprints/hta.html ER - TY - JOUR A1 - Bär, Christian A1 - Mazzeo, Rafe T1 - Manifolds with many Rarita-Schwinger fields JF - Communications in mathematical physics N2 - The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kahler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension. Y1 - 2021 U6 - https://doi.org/10.1007/s00220-021-04030-0 SN - 0010-3616 SN - 1432-0916 VL - 384 IS - 1 SP - 533 EP - 548 PB - Springer CY - Berlin ER - TY - JOUR A1 - Bär, Christian A1 - Hanke, Bernhard T1 - Local flexibility for open partial differential relations JF - Communications on pure and applied mathematics / issued by the Courant Institute of Mathematical Sciences, New York Univ. N2 - We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry. The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K is an element of Double-struck capital R every manifold of dimension at least 2 carries a complete C-1,C- 1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course, this is impossible for C-2-metrics in general. Y1 - 2021 U6 - https://doi.org/10.1002/cpa.21982 SN - 0010-3640 SN - 1097-0312 VL - 75 IS - 6 SP - 1377 EP - 1415 PB - Wiley CY - Hoboken ER - TY - JOUR A1 - Bär, Christian A1 - Gouduchon, Paul A1 - Moroianu, Andrei T1 - Generalized Cylinders in Semi-Riemannian and Spin Geometry N2 - 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. Y1 - 2005 UR - http://xxx.uni-augsburg.de/abs/math.DG/0303095 ER -