The spacelike-characteristic Cauchy problem of general relativity in low regularity

TL;DR

This paper establishes local existence results for the Einstein vacuum equations with low regularity initial data on spacelike and null hypersurfaces, extending previous theories to less smooth settings.

Contribution

It proves the existence of solutions under low regularity curvature bounds, utilizing advanced geometric analysis techniques and extending prior results to broader initial data classes.

Findings

Existence of solutions with low regularity initial data.

Control of solution lifespan by $L^2$ curvature bounds.

Application of low regularity geometric analysis methods.

Abstract

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface $\Sigma \simeq \overline{B(0,1)} \subset \mathbb{R}^3$ and the outgoing null hypersurface $\mathcal{H}$ emanating from $\partial \Sigma$, the time of existence of a solution to the Einstein vacuum equations is controlled by low regularity bounds on the initial data at the level of curvature in $L^2$. The proof uses the bounded $L^2$ curvature theorem by Klainerman, Szeftel and Rodnianski, the extension procedure for the constraint equations by Czimek, Cheeger-Gromov theory in low regularity developed by Czimek, the canonical foliation on null hypersurfaces in low regularity by Czimek and Graf, and global elliptic estimates for spacelike maximal hypersurfaces.