Year: 2026
Type: preprint
Abstract: We prove the Hawking and Penrose singularity theorems for Lorentz--Finsler structures of mixed horizontal/vertical regularity $C^{(1,2)}$ in the $C^1$ weighted case and $C^{(1,3)}$ in the unweighted case. At this regularity, geodesics need not be uniquely determined by their initial conditions, and the (weighted) Ricci curvature must be interpreted distributionally. Our approach relies on a two-stage approximation procedure: first, a homogeneity-preserving convolution, which is of independent interest in the broader semi-Riemann--Finsler setting, and second, a causality-adapted approximation à la Chruściel--Grant. In the special case of $C^1$ Lorentzian metrics, our results extend the $C^1$ singularity theorems of Graf to the $C^1$ weighted case.
Source: arXiv (Cornell University)
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: 最適輸送理論の深化－数理情報の融合研究－, Analysis on metric measure spaces by optimal transport theory and Markov processes, G166807448, New developments in gradient flow theory on metric spaces, Lorentzian Metric Geometry and Optimal Transport, Geometric Analysis and Applied Mathematics Integrated through Mathematical Optimization, G3886868163, G5820727336, Singularities and rigidity in non-smooth spacetimes, G7525619771, Lorentzian length spaces, G9000264761
OpenAlex ID: W7221131845