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Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

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Abstract

Given a Riemannian 3-ball \(({\bar{B}}, g)\) of nonnegative 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.

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Authors and Affiliations

  1. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11790, USA

    Michael T. Anderson

  2. Department of Mathematics, Union College, Schenectady, NY, 12308, USA

    Jeffrey L. Jauregui

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  1. Michael T. Anderson
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  2. Jeffrey L. Jauregui
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Correspondence to Michael T. Anderson.

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Communicated by Mihalis Dafermos.

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Anderson, M.T., Jauregui, J.L. Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures. Ann. Henri Poincaré 20, 1651–1698 (2019). https://doi.org/10.1007/s00023-019-00786-3

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