The localised bounded $L^2$-curvature theorem

TL;DR

This paper establishes a localized version of the bounded $L^2$-curvature theorem for Einstein vacuum equations, showing that the existence time depends on localized curvature bounds and volume radius, using a covering argument.

Contribution

It introduces a localized bounded $L^2$-curvature theorem for Einstein vacuum equations, extending previous global results to local settings with boundary considerations.

Findings

Existence time depends on localized curvature bounds and volume radius.

Reduction from large to small data via scaling and covering arguments.

Utilizes previous work and black-box theorems for proof.

Abstract

In this paper, we prove a localised version of the bounded $L^2$-curvature theorem of Klainerman-Rodnianski-Szeftel. More precisely, we consider initial data for the Einstein vacuum equations posed on a compact spacelike hypersurface $\Sigma$ with boundary, and show that the time of existence of a classical solution depends only on an $L^2$-bound on the Ricci curvature, an $L^4$-bound on the second fundamental form of $\partial \Sigma \subset \Sigma$, an $H^1$-bound on the second fundamental form, and a lower bound on the volume radius at scale $1$ of $\Sigma$. Our localisation is achieved by first proving a localised bounded $L^2$-curvature theorem for small data posed on $B(0,1)$, and then using the scaling of the Einstein equations and a low regularity covering argument on $\Sigma$ to reduce from large data on $\Sigma$ to small data on $B(0,1)$. The proof uses the author's previous work, and the bounded $L^2$-curvature theorem as black boxes.