Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions

McCormick, Stephen

doi:10.2140/pjm.2020.304.629

Cited by 7 publications

(11 citation statements)

References 22 publications

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“…However, this does not directly imply lower semi-continuity of the Bartnik mass: the main difficulty is that it is not known that "close" Bartnik data necessarily have "close" competitors for near-minimal mass extensions. Note that the recent work of McCormick [42] on the continuity of the Bartnik mass does not apply here -the regions we consider will not generally satisfy the required convexity condition in [42,Theorem 5.1].…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…Both of these versions have previously appeared in the literature. For further discussion on the numerous variations in the definition of Bartnik mass, and some progress on reconciling them, see [32], [42]. These three definitions, based on (1.1)-(1.3), all require a precise choice among the various possible definitions of horizon.…”

Section: Introductionmentioning

confidence: 99%

“…This result strongly suggests the two definitions of Bartnik mass based on (1.2) and (1.3) are equivalent and also very likely equivalent to (1.1), cf. Remark 2.9 and [42], [32]. Henceforth, we adopt (1.4) as the definition of the Bartnik mass.…”

Section: Introductionmentioning

confidence: 99%

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Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

Anderson

Jauregui

2019

Ann. Henri Poincaré

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Given a Riemannian 3-ball (B, g) of non-negative 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.

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Section: Proof Of Theorem 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

Anderson

Jauregui

2019

Ann. Henri Poincaré

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“…The continuity and differentiability of the Bartnik mass is another challenging question that remains to be rigorously analyzed. In the time-symmetric setting, partial results on the continuity of the Barntnik mass has been given by the first-named author [15]. In the general case, existence and smooth dependence of stationary vacuum extensions of boundary data close to a round sphere in the Minkowski spacetime R 3,1 has been recently obtained by Z.…”

Section: )mentioning

confidence: 99%

On the evolution of the spacetime Bartnik mass

McCormick

Miao

2019

Pure and Applied Mathematics Quarterly

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It is conjectured that the full (spacetime) Bartnik mass of a surface Σ is realised as the ADM mass of some stationary asymptotically flat manifold with boundary data prescribed by Σ. Assuming this holds true for a 1-parameter family of surfaces Σ t evolving in an initial data set with the dominant energy condition, we compute an expression for the derivative of the Bartnik mass along these surfaces. An immediate consequence of this formula is that the Bartnik mass of Σ t is monotone non-decreasing whenever Σ t flows outward.It is our pleasure to dedicate this paper to Robert Bartnik on the occasion of his 60th birthday.

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“…Shortly before this paper was posted to the arXiv, S. McCormick posted [25] to the arXiv, which is on a very similar topic but uses different techniques. In particular, Theorem 3.3 therein corresponds with Theorem 2 above.…”

Section: Introductionmentioning

confidence: 99%

Smoothing the Bartnik boundary conditions and other results on Bartnik’s quasi-local mass

Jauregui

2019

Journal of Geometry and Physics

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Quite a number of distinct versions of Bartnik's definition of quasi-local mass appear in the literature, and it is not a priori clear that any of them produce the same value in general. In this paper we make progress on reconciling these definitions. The source of discrepancies is two-fold: the choice of boundary conditions (of which there are three variants) and the non-degeneracy or "no-horizon" condition (at least six variants). To address the boundary conditions, we show that given a 3-dimensional region Ω of nonnegative scalar curvature (R ≥ 0) extended in a Lipschitz fashion across ∂Ω to an asymptotically flat 3-manifold with R ≥ 0 (also holding distributionally along ∂Ω), there exists a smoothing, arbitrarily small in C 0 norm, such that R ≥ 0 and the geometry of Ω are preserved, and the ADM mass changes only by a small amount. With this we are able to show that the three boundary conditions yield equivalent Bartnik masses for two reasonable non-degeneracy conditions. We also discuss subtleties pertaining to the various non-degeneracy conditions and produce a nontrivial inequality between a no-horizon version of the Bartnik mass and Bray's replacement of this with the outward-minimizing condition. arXiv:1806.08348v1 [math.DG]

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Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions

Cited by 7 publications

References 22 publications

Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

On the evolution of the spacetime Bartnik mass

Smoothing the Bartnik boundary conditions and other results on Bartnik’s quasi-local mass

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