2021/07/18 by Orville Damaschke, Damaschke, Orville
Mathematics · #46L10 #53C50 #58J20 #58J40 (Secondary) #58J45 (Primary) 35L03 #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2107.08532
openalex publication_date 2021/07/18 · openalex created_date 2021/08/02 · openalex updated_date 2026/07/28
Let M be a globally hyperbolic manifold with complete spacelike Cauchy hypersurface Σ⊂ 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.