Embeddings, immersions and the Bartnik quasi-local mass conjectures

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.

38 pages, 4 figures

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Cited by in corpus (3)

• Outer entropy = Bartnik-Bray inner mass, and the gravitational ant conjecture
• New asymptotically flat static vacuum metrics with near Euclidean boundary data
• Static vacuum extensions with prescribed Bartnik boundary data near a general static vacuum metric