Quasi-local Mass Integrals and the Total Mass

In this talk, we will describe known results on how to evaluate the ADM mass of an asymptotically flat manifold and the mass integral of an asymptotically hyperbolic manifold. We then apply the results to relate Brown-York quasi-local mass of coordinate spheres and the ADM mass of an asymptotically flat manifold. We will also relate some quasi-local mass integrals of coordinate spheres and the total mass integral of an asymptotically hyperbolic manifold. We will also discuss some functionals on compact manifolds with boundary which are motivated by these considerations.