Local existence and uniqueness for exterior static vacuum Einstein metrics

TL;DR

This paper proves the local existence and uniqueness of asymptotically flat solutions to the static vacuum Einstein equations on exterior domains with prescribed boundary data, near the standard flat boundary conditions.

Contribution

It establishes a local existence and uniqueness result for static vacuum Einstein metrics with boundary data close to the Euclidean case.

Findings

Existence of unique AF solutions near standard flat boundary data.

Solutions are close to the standard Euclidean solution.

Results apply to prescribed boundary metric and mean curvature.

Abstract

We study solutions to the static vacuum Einstein equations on exterior domains with prescribed metric and mean curvature on the inner boundary. It is proved that for any such boundary data near the standard round boundary data in Euclidean space, there exists a unique AF solution to the static vacuum equations realizing the boundary data, which is close to the standard flat solution.