Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature

TL;DR

This paper reviews recent advances in understanding the local geometry of Lorentzian manifolds with bounded curvature, focusing on injectivity radius estimates and canonical foliations using minimal regularity assumptions.

Contribution

It introduces new methods to estimate the injectivity radius and construct canonical foliations with only a curvature sup-norm bound, improving upon previous derivative-based bounds.

Findings

Injectivity radius estimates near observers.

Existence of local canonical CMC foliations.

Construction of spatially harmonic coordinates.

Abstract

We review recent work on the local geometry and optimal regularity of Lorentzian manifolds with bounded curvature. Our main results provide an estimate of the injectivity radius of an observer, and a local canonical foliations by CMC (Constant Mean Curvature) hypersurfaces, together with spatially harmonic coordinates. In contrast with earlier results based on a global bound for derivatives of the curvature, our method requires only a sup-norm bound on the curvature near the given observer.