A short note on the Bartnik mass

In this note we use a localized deformation construction for the Einstein constraint equations to obtain a variational condition that must be satisfied by a suitable Bartnik quasi-local mass minimizer in the general non-time-symmetric setting.

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Many thanks to Robert Bartnik for inspiring so many of us over the years. We warmly thank Lan-Hsuan Huang for her encouragement and discussion, as well as Jeff Jauregui, Stephen McCormick and Pengzi Miao for discussing aspects of this note. We also thank Piotr Chruściel and Shing-Tung Yau for their encouragement, and two referees for their careful reading and thoughtful comments. The work was partly supported by an R.K. Mellon Fellowship at Lafayette College.

Received 30 July 2020

Accepted 21 November 2019

Published 2 January 2020