Uniqueness of static vacuum Einstein metrics and the Bartnik quasi-local mass

TL;DR

This paper investigates the uniqueness of static vacuum Einstein solutions with specific boundary data, revealing cases where uniqueness fails and exploring implications for the Bartnik quasi-local mass, along with a variational characterization.

Contribution

It constructs examples showing non-uniqueness in static vacuum Einstein solutions and discusses the impact on Bartnik quasi-local mass, including a variational approach.

Findings

Constructed classes where uniqueness fails

Implications for Bartnik quasi-local mass analyzed

Provided a variational characterization of boundary data

Abstract

We analyse the issue of uniqueness of solutions of the static vacuum Einstein equations with prescribed geometric or Bartnik boundary data. Large classes of examples are constructed where uniqueness fails. We then discuss the implications of this behavior for the Bartnik quasi-local mass. A variational characterization of Bartnik boundary data is also given.