Lipschitz continuity of the cut time for globally hyperbolic spacetimes

We prove that, for a fixed point in a globally hyperbolic spacetime, the focalization time is locally Lipschitz continuous on the open subset of the future causal cone where it is finite. We also show that the cut time is locally Lipschitz continuous in a neighborhood of any timelike tangent vector whose associated geodesic is defined at least up to (and including) its cut time. Furthermore, we derive quantitative estimates for the Lipschitz constant near the null cone and provide a criterion ensuring that the Lipschitz property extends to null directions. As a consequence, we show that the cut locus of a point has Hausdorff codimension at least $1$. These results extend classical Lipschitz continuity results for complete Riemannian manifolds due to Itoh-Tanaka and Li-Nirenberg. Our approach follows the method of Itoh-Tanaka, suitably adapted to the Lorentzian setting.