Local canonical foliations of Lorentzian manifolds with bounded curvature

TL;DR

This paper constructs canonical foliations by constant mean curvature hypersurfaces in Lorentzian manifolds with bounded curvature, enabling improved regularity analysis of the metric.

Contribution

It introduces a method to create CMC foliations under minimal curvature and injectivity bounds, leading to optimal regularity coordinates in Lorentzian geometry.

Findings

Established existence of CMC foliations under bounded curvature.

Introduced CMC-harmonic coordinates with optimal regularity.

Provided tools for analyzing Lorentzian manifolds with limited regularity.

Abstract

We consider pointed Lorentzian manifolds and construct "canonical" foliations by constant mean curvature (CMC) hypersurfaces. Our result assumes a uniform bound on the local sup-norm of the curvature of the manifold and on its local injectivity radius, only. The prescribed curvature problem under consideration is a nonlinear elliptic equation whose coefficients have limited regularity. The CMC foliation allows us to introduce CMC-harmonic coordinates, in which the coefficients of the Lorentzian metric have optimal regularity.