On the Bartnik extension problem for the static vacuum Einstein equations

TL;DR

This paper investigates the existence of asymptotically flat solutions to the static vacuum Einstein equations with specific boundary data, partially addressing Bartnik's conjecture and its relation to quasi-local mass.

Contribution

It develops a framework for static vacuum extensions with prescribed boundary data and provides partial existence results, advancing understanding of Bartnik's conjecture.

Findings

Partial existence of static vacuum extensions established

Framework for boundary data prescribed solutions developed

Connections to Bartnik's quasi-local mass clarified

Abstract

We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations with prescribed boundary data consisting of the induced metric and mean curvature on a 2-sphere. A partial existence result is obtained, giving a partial resolution of a conjecture of Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is closely related to Bartnik's definition of quasi-local mass.