Mass of asymptotically flat $3$-manifolds with boundary

TL;DR

This paper investigates the mass of asymptotically flat 3-manifolds with boundary, deriving a new mass formula and conditions for positivity, with applications to quasi-local mass and Brown-York mass convergence.

Contribution

It introduces a new mass formula for such manifolds and provides sufficient conditions for mass positivity, extending previous methods.

Findings

Derived a mass formula for manifolds with boundary.

Established conditions guaranteeing mass positivity.

Proved convergence of Brown-York mass for large surfaces.

Abstract

We study the mass of asymptotically flat $3$-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern. 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 $\infty$ which include the scaling of any fixed coordinate-convex surface.