Abstract

Static manifolds are Riemannian manifolds of vanishing scalar curvature which give rise to equilibrium solutions of the Einstein equations in general relativity. In addition to playing a central role in the initial value problem of relativity, these manifolds are also important in definitions of mass in GR. While the Arnowitt-Deser-Misner (ADM) mass provides a suitable notion of total mass of an asymptotically flat, time-symmetric initial data set, defining the mass of a bounded open region in this data set is far more elusive.

Originally proposed by Robert Bartnik in 1986, the Bartnik mass is perhaps the most natural definition of quasi-local mass in GR. Given a compact Riemannian manifold   with a spherical boundary component , the Bartnik mass of   is conjectured to equal the ADM mass of an asymptotically flat static manifold which induces the same metric and mean curvature over as . This presents a challenge to differential geometers: show that a unique static extension always exists for any choice of positively-curved boundary metric and positive mean curvature function in order to verify the validity of the Bartnik mass.

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