The Bartnik quasi-local mass conjectures

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.

Keywords

quasi-local mass, elliptic boundary value problems, non-linear Fredholm maps

2010 Mathematics Subject Classification

Primary 58J32, 83C05. Secondary 35J25, 58D27.

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In Memory of Robert Bartnik

Received 1 August 2023

Accepted 31 January 2024

Published 17 July 2024