Embeddings, immersions and the Bartnik quasi-local mass conjectures

TL;DR

This paper investigates Bartnik's conjectures on minimal mass extensions of Riemannian 3-balls with non-negative scalar curvature, proving the uniqueness of static vacuum solutions when they exist, but showing minimal mass extensions do not always exist.

Contribution

It proves that mass-minimizers, if they exist, are static vacuum solutions, and demonstrates that such minimal extensions are not always obtainable for all bodies.

Findings

Mass-minimizers are static vacuum solutions when they exist.

Minimal mass extensions do not exist for a large class of bodies.

The first Bartnik conjecture is false in general.

Abstract

Given a Riemannian 3-ball $(\bar B, g)$ of non-negative scalar curvature, Bartnik conjectured that $(\bar B, g)$ admits an asymptotically flat (AF) extension (without horizons) of the least possible ADM mass, and that such a mass-minimizer is an AF solution to the static vacuum Einstein equations, uniquely determined by natural geometric conditions on the boundary data of $(\bar B, g)$. We prove the validity of the second statement, i.e.~such mass-minimizers, if they exist, are indeed AF solutions of the static vacuum equations. On the other hand, we prove that the first statement is not true in general; there is a rather large class of bodies $(\bar B, g)$ for which a minimal mass extension does not exist.