Large-sphere and small-sphere limits of the Brown-York mass

TL;DR

This paper investigates the asymptotic behavior of Brown-York and Hawking masses of spheres in asymptotically flat manifolds, revealing their limits and geometric implications.

Contribution

It provides new insights into the limits and expansions of Brown-York and Hawking masses, connecting them with geometric properties of the manifold.

Findings

Limits of Brown-York mass for coordinate spheres are characterized.

Expansions of Brown-York and Hawking masses for geodesic spheres are derived.

Geometric consequences of mass behaviors are discussed.

Abstract

In this paper, we will study the limiting behavior of the Brown-York mass of the coordinate spheres in an asymptotically flat manifold. Limiting behaviors of volumes of regions related to coordinate spheres are also obtained, including a discussion on the isoperimetric mass introduced by Huisken \cite{Huisken}. We will also study expansions of the Brown-York mass and the Hawking mass of geodesic spheres with center at a fixed point $p$ of a three manifold. Some geometric consequences will be derived.