On a spacetime positive mass theorem with corners

Abstract

In this paper we consider the positive mass theorem for general initial data sets satisfying the dominant energy condition which are singular across a piecewise smooth surface. We find jump conditions on the metric and second fundamental form which are sufficient for the positivity of the total spacetime mass. Our method extends that of [30] to the singular case (which we refer to as initial data sets with corners) using some ideas from [31]. As such we give an integral lower bound on the spacetime mass and we characterise the case of zero mass. Our approach also leads to a new notion of quasilocal mass which we show to be positive, extending the work of [54] to the spacetime case. Moreover, we give sufficient conditions under which spacetime Bartnik data sets cannot admit a fill-in satisfying the dominant energy condition. This generalises the work of [58] and [57] to the spacetime setting.

Nonexistence of DEC spin fill-ins

“…This question was tackled by Tsang in [Tsa22] where several partial results are proved. We remark that if k = 0 then α = 0 and h = 0, and the DEC gives the nonnegativity of the scalar curvature so that a DEC fill-in of (Σ, γ, H) corresponds to a NNSC fill-in.…”