Mass of asymptotically flat $3$-manifolds with boundary

Abstract

We study the mass of asymptotically flat 3-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern [5]. More precisely, we derive a mass formula on the union of an asymptotically flat manifold and fill-ins of its boundary, and give new sufficient conditions guaranteeing the positivity of the mass. Motivation to such consideration comes from studying the quasi-local mass of the boundary surface. If the boundary isometrically embeds in the Euclidean space, we apply the formula to obtain convergence of the Brown-York mass along large surfaces tending to ∞ which include the scaling of any fixed coordinate-convex surface.Proposition 1.1. Let (M, g) be an asymptotically flat 3-manifold with boundary Σ. Let u be a harmonic function on M which is asymptotic to one of the asymptotically flat coordinate functions at infinity. Let Σ t = u −1 (t) be the level set of u and ν be the infinity pointing unit P. Miao acknowledges the support of NSF Grant DMS-1906423.

On a spacetime positive mass theorem with corners

“…We here assume that |∇u| = 0 for the simplicity of presentation. For the full generality, one should first consider |∇u| 2 + δ 2 for δ > 0 and then take limit as δ → 0 (see [61], [13], [30], [31] Remark 3.3). It suffices to verify the divergence theorem such that the following holds.…”

“…We will express the boundary terms of Lemma 5.1 explicitly for spacetime harmonic functions on manifolds with boundary. Note that we have to make use of the fact that ∆u = −K|∇u| instead of 0 in [31].…”

“…If k ≡ 0, for non-smooth initial data sets, the problem has also been extensively studied ( [35], [36], [38], [41], [44], [54], [56]). By generalising the integral formula in [61] and [12], it is shown in [31] that, if the metric is not smooth across a closed hypersurface (corner), then an integral of the mean curvature difference would determine a lower bound of the mass. For the spacetime case, as motivated by the Hamiltonian formulation ( [14], [28]), the positive mass theorem with non-compact boundary has been studied in [4].…”

“…This also suggests conditions to be imposed on the corner for the positive mass theorem. Together, with the approach in [31], we generalise the integral formula in [30] and get the following results.…”

“…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.…”