Einstein Constraints on Asymptotically Euclidean Manifolds

TL;DR

This paper extends the constructive methods for solving Einstein constraint equations on asymptotically Euclidean manifolds, including cases with discontinuous sources and non-constant mean curvature, advancing mathematical understanding in general relativity.

Contribution

It introduces new existence proofs for Einstein constraints on asymptotically Euclidean manifolds, handling both scaled and unscaled sources, and addresses discontinuous sources and variable mean curvature.

Findings

Established existence of solutions with scaled and unscaled sources

Extended methods to include discontinuous sources

Derived new results for non-constant mean curvature cases

Abstract

We consider the Einstein constraints on asymptotically euclidean manifolds $M$ of dimension $n \geq 3$ with sources of both scaled and unscaled types. We extend to asymptotically euclidean manifolds the constructive method of proof of existence. We also treat discontinuous scaled sources. In the last section we obtain new results in the case of non-constant mean curvature.