Global characteristic problem for Einstein vacuum equations with small initial data: (I) The initial data constraints

TL;DR

This paper details the construction of initial data satisfying constraints and decay conditions for Einstein vacuum equations, laying the groundwork for a subsequent proof of global existence in a characteristic setting.

Contribution

It introduces a method to prescribe initial data with constraints, smallness, regularity, and decay for Einstein vacuum equations in a characteristic framework.

Findings

Constructed initial data satisfying constraints and decay conditions.

Outlined the approach for proving global existence in a subsequent work.

Extended techniques from non-characteristic to characteristic problems.

Abstract

We show how to prescribe the initial data of a characteristic problem satisfying the costraints, the smallness, the regularity and the asymptotic decay suitable to prove a global existence result. In this paper, the first of two, we show in detail the construction of the initial data and give a sketch of the existence result. This proof, which mimicks the analogous one for the non characteristic problem in [Kl-Ni], will be the content of a subsequent paper.