The maximal existence region of the analytic solutions for a class of characteristic Einstein vacuum equations

TL;DR

This paper demonstrates that the existence region for analytic solutions to certain characteristic Einstein vacuum equations extends beyond classical results, depending on Sobolev norms, and can be global for small initial data.

Contribution

It establishes a larger existence region for analytic solutions of characteristic Einstein vacuum equations using hyperbolicity and Sobolev norms, improving upon previous local results.

Findings

Existence region exceeds Cauchy-Kowalevski bounds due to hyperbolicity.

Small initial data lead to global analytic solutions.

Dependence on Sobolev norms of initial data.

Abstract

The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region larger than the one provided by the Cauchy-Kowalevski theorem, due to the intrinsic hyperbolicity of the Einstein equations. The magnitude of this region depends only on suitable $H_s$ Sobolev norms of the initial data for a fixed $s\leq 7$ and if the initial data are sufficiently small the analytic solution is global. In a previous paper, hereafter "I", we have described a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering and a local solution in a suitable "double null cone gauge" characterized by the use of a double null cone foliation of the spacetime.