arXiv subjects

Orville Damaschke

Publications and source records attributed to Orville Damaschke.

3 recordsLinked to original sources

arXiv2024

$L^2$-Gamma-Fredholmness for spacetimes

Let $M$ be a temporal compact globally hyperbolic manifold with Cauchy hypersurface $Σ$ which is a Galois covering with respect to a discrete group $Γ$ of automorphisms such that the quotient $Σ/Γ$ is compact without boundary. We will show Fredholmness of a (spatial) $Γ$-invariant Lorentzian Dirac operator under (anti) Atiyah-Patodi-Singer boundary conditions in the von Neumann sense, known as ($L^2$-)$Γ$-Fredholmness. The results already have been published for a simpler setting in arXiv:2107.08532. This version is only focused on the Fredholmness part, based on previous results from arXiv:2409.17344, and also generalised (anti) Atiyah-Patodi-Singer boundary conditions are going to be considered as well.

math.DG↗

arXiv2024

Cauchy Problem for the Dirac operator on spatially non-compact spacetimes

Let $M$ be a globally hyperbolic manifold with complete spacelike Cauchy hypersurface $Σ$. We prove well-posedness of the Cauchy problem for the Dirac operator on globally hyperbolic manifolds with complete Cauchy hypersurfaces. This result is needed as preparation in showing a Fredholmness result in the manner, provided by Bär and Strohmaier, for certain non-compact Cauchy hypersurfaces in future work. The results already have been published for a simpler setting in arxiv:2107.08532. This version is only focused on the Cauchy problem for a slightly modified setting.

math.DG↗

arXiv2021

Atiyah-Singer Dirac Operator on spacetimes with non-compact Cauchy hypersurface

Let $M$ be a globally hyperbolic manifold with complete spacelike Cauchy hypersurface $Σ\subset M$. Building on past and recent works of Bär and Strohmaier, we extend their Fredholm result of the Atiyah-Singer Dirac operator on compact Lorentzian spaces to the case, where $M$ is diffeomorphic to a product of $Σ$ with a compact time intervall and the hypersurface is a Galois covering with respect to a group $Γ$. We follow the first approach of both authors in this extended setting, where a well-posedness result of the Cauchy problem for the Dirac operator on non-compact manifolds is needed in preparation. After employing von Neumann algebras and further ingredients for Galois coverings, the well-posedness result is specified for the setting of interest, which leads to $Γ$-Fredholmness of the Dirac operator under APS boundary conditions.

math.DG↗