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Annales Henri Poincaré

, Volume 20, Issue 5, pp 1651–1698 | Cite as

Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

  • Michael T. AndersonEmail author
  • Jeffrey L. Jauregui
Article
  • 32 Downloads

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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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA
  2. 2.Department of MathematicsUnion CollegeSchenectadyUSA

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