Lorentzian Gromov–Hausdorff theory and finiteness results

Cheeger–Gromov finiteness results, asserting that there are only finitely many diffeomorphism types of manifolds satisfying certain geometric bounds, feature among the most prominent results in Riemannian geometry. To transplant those into Lorentzian geometry, one could use a functor between a Lorentzian and a Riemannian category, which, however, can be shown not to exist if the former contains Minkowski space and its isometries. Here, we construct a functor from a restricted category of Lorentzian manifolds-with-boundary (regions between two Cauchy surfaces) to a category of Riemannian manifolds-with-boundary that preserves geometric bounds and obtain, as a corollary, the first known Lorentzian Cheeger–Gromov type finiteness result.