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

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.