In the reviewed paper the author generalizes certain volume comparison theorems for smooth Riemannian or Lorentzian manifolds to metrics that are only \(C^{1,1}\). For a (semi-)Riemannnian metric the class \(C^{1,1}\) (locally Lipschitz continuous first derivatives) is the lowest differentiability class of the metric where one still has local existence and uniqueness of solutions of the geodesic equation.
First the author studies Riemannian manifolds with \(C^{1,1}\)-metrics with a lower bound on the Ricci curvature and shows a \(C^{1,1}\) version of the Bishop-Gromov volume comparison theorem. In the Lorentzian case, she first gives the definition of the cosmological comparison condition, then shows the existence of suitable approximating metrics and that for \(C^{1,1}\)-metrics the cut locus still has measure zero. Next, the author defines comparison spacetimes as Robertson-Walker spacetimes with constant Ricci curvature and studies their dependence on the curvature quantities \(\kappa\) and \(\beta\). Finally, she proves a volume comparison theorem and shows some applications of this.