Communications in Analysis and Geometry

Volume 31 (2023)

Number 7

Mass of asymptotically flat $3$-manifolds with boundary

Pages: 1611 – 1653

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n7.a1

Authors

Sven Hirsch (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Pengzi Miao (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)

Tin-Yau Tsang (Department of Mathematics, University of California, Irvine, Calif., U.S.A.)

Abstract

We study the mass of asymptotically flat $3$-manifolds with boundary using the method of Bray–Kazaras–Khuri–Stern $\href{https://doi.org/10.48550/arXiv.1911.06754}{[6]}$. 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.

Received 18 September 2020

Accepted 23 June 2021

Published 10 August 2024