The Bartnik quasi-local mass conjectures

TL;DR

This paper examines Bartnik's quasi-local mass conjectures, providing a framework to analyze the static vacuum extension conjecture, proving its general falsehood while highlighting its significance and open questions.

Contribution

It introduces a framework for analyzing Bartnik's conjecture and demonstrates its limitations, advancing understanding of quasi-local mass in general relativity.

Findings

The static vacuum extension conjecture is not true in general.

A new analytical framework for studying Bartnik's conjecture is proposed.

Open problems remain regarding the conjecture's domain of validity.

Abstract

This paper is a tribute to Robert Bartnik and his work and conjectures on quasi-local mass. We present a framework in which to clearly analyse Bartnik's static vacuum extension conjecture. While we prove that this conjecture is not true in general, it remains a fundamental open problem to understand the realm of its validity.