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Given a Riemannian 3-ball (B ¯ , g) of nonnegative scalar curvature, Bartnik conjectured that (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 (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 (B ¯ , g) for which a minimal mass extension does not exist. [ABSTRACT FROM AUTHOR]

Abstract: Corrections are given to some of the proofs of the paper mentioned in the title. [Copyright &y& Elsevier]

Abstract: Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss–Codazzi constraint equations under deformations are also discussed. [Copyright &y& Elsevier]

It is shown that there is a unique Yamabe representative for a generic set of conformal classes in the space of metrics on any manifold. At such classes, the scalar curvature functional is shown to be differentiable on the space of Yamabe metrics. In addition, some sufficient conditions are given which imply that a Yamabe metric of locally maximal scalar curvature is necessarily Einstein. [ABSTRACT FROM AUTHOR]