## Prof. Dr. Christian Bär

### Deputy Speaker, Member of Programme committee, Project leader

**Professor**

Universität Potsdam

E-mail: baer(at)math.uni-potsdam.de

Telephone: +49 331 977-1348

Homepage: https://www.math.uni-potsdam.de/baer/

## Publications within SPP2026

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.

Journal | Amer. J. Math. |

Publisher | John Hopkins Univ. Press |

Volume | 141 (5) |

Pages | 1421-1455 |

Link to preprint version | |

Link to published version |

**Related project(s):****5**Index theory on Lorentzian manifolds

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.

We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We provide examples which are conveniently treated by our methods.

**Related project(s):****5**Index theory on Lorentzian manifolds

On a compact globally hyperbolic Lorentzian spin manifold with smooth spacelike Cauchy boundary the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah-Patodi-Singer boundary conditions are imposed. In this paper we investigate to what extent these boundary conditions can be replaced by more general ones and how the index then changes. There are some differences to the classical case of the elliptic Dirac operator on a Riemannian manifold with boundary.

Publisher | Oxford University Press |

Book | A. Dancer, J.E. Andersen, O. García-Prada (eds.): Geometry and Physics |

Volume | 1 |

Pages | 3-18 |

Link to preprint version | |

Link to published version |

**Related project(s):****5**Index theory on Lorentzian manifolds

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 which have very restrictive local properties an open dense subsets. This shows, for instance, that given any *K*∈R every manifold of dimension at least two carries a complete *C^*1,1-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.

**Related project(s):****5**Index theory on Lorentzian manifolds**15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on black-hole spacetimes (in particular, on the Kerr solution), the index theorem for hyperbolic Dirac operators and properties of the class of Green-hyperbolic operators.

Publisher | de Gruyter |

Book | J. Brüning, M. Staudacher (Eds.): Space - Time - Matter |

Pages | 324-348 |

Link to preprint version | |

Link to published version |

**Related project(s):****5**Index theory on Lorentzian manifolds

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 in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the *η*-invariant of the Cauchy hypersurfaces.

Journal | Commun. Math. Phys. |

Publisher | Springer |

Volume | 347 |

Pages | 703-721 |

Link to preprint version | |

Link to published version |

**Related project(s):****5**Index theory on Lorentzian manifolds