Some lower estimates of ADM mass and Brown-York mass

We give some lower estimates of the ADM mass of an asymptotically flat (AF) Riemannian manifold without assuming that the scalar curvature of the manifold is nonnegative. Some sufficient conditions for an AF manifold to have nonnegative ADM mass are obtained. We also give some lower estimates of the Brown-York mass of a compact three manifold with smooth boundary. From these estimates, we generalize some previous results of the authors.