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Title: A note on mass-minimising extensions
Contributor(s): McCormick, Stephen  (author)orcid 
Publication Date: 2015
Open Access: Yes
DOI: 10.1007/s10714-015-1993-2Open Access Link
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Abstract: A conjecture related to the Bartnik quasilocal mass, is that the infimum of the ADM energy, over an appropriate space of extensions to a compact 3-manifold with boundary, is realised by a static metric. It was shown by Corvino (Commun Math Phys 214(1):137-189, 2000) that if the infimum is indeed achieved, then it is achieved by a static metric; however, the more difficult question of whether or not the infimum is achieved, is still an open problem. Bartnik (Commun Anal Geom 13(5):845-885, 2005) then proved that critical points of the ADM mass, over the space of solutions to the Einstein constraints on an asymptotically flat manifold without boundary, correspond to stationary solutions. In that article, he stated that it should be possible to use a similar construction to provide a more natural proof of Corvino's result. In the first part of this note, we discuss the required modifications to Bartnik's argument to adapt it to include a boundary. Assuming that certain results concerning a Hilbert manifold structure for the space of solutions carry over to the case considered here, we then demonstrate how Bartnik's proof can be modified to consider the simpler case of scalar-flat extensions and obtain Corvino's result. In the second part of this note, we consider a space of extensions in a fixed conformal class. Sufficient conditions are given to ensure that the infimum is realised within this class.
Publication Type: Journal Article
Source of Publication: General Relativity and Gravitation, 47(12), p. 1-14
Publisher: Springer New York LLC
Place of Publication: New York, United States of America
ISSN: 1572-9532
Field of Research (FOR): 010504 Mathematical Aspects of General Relativity
010110 Partial Differential Equations
Peer Reviewed: Yes
HERDC Category Description: C1 Refereed Article in a Scholarly Journal
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