Abstract. Let R be a constant. Let M R γ be the space of smooth metrics g on a given compact manifold Ω n (n ≥ 3) with smooth boundary Σ such that g has constant scalar curvature R and g| Σ is a fixed metric γ on Σ. Let V (g) be the volume of g ∈ M R γ . In this work, we classify all Einstein or conformally flat metrics which are critical points of V (·) in M R γ .

We prove directly without using a density theorem that (i) the ADM mass defined in the usual way on an asymptotically flat manifold is equal to the mass defined intrinsically using the Ricci tensor; (ii) the Hamiltonian formulation of center of mass and the center of mass defined intrinsically using the Ricci tensor are the same.

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On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown-York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.

Abstract. We discuss some geometric problems related to the definitions of quasilocal mass proposed by [6] and Liu-Yau [13] [14]. Our discussion consists of three parts. In the first part, we propose a new variational problem on compact manifolds with boundary, which is motivated by the study of Brown-York mass. We prove that critical points of this variation problem are exactly static metrics. In the second part, we derive a derivative formula for the Brown-York mass of a smooth family of closed 2 dimensional surfaces evolving in an ambient three dimensional manifold. As an interesting by-product, we are able to write the ADM mass [1] of an asymptotically flat 3-manifold as the sum of the Brown-York mass of a coordinate sphere S r and an integral of the scalar curvature plus a geometrically constructed function Φ(x) in the asymptotic region outside S r . In the third part, we prove that for any closed, spacelike, 2-surface Σ in the Minkowski space R 3,1 for which the Liu-Yau mass is defined, if Σ bounds a compact spacelike hypersurface in R 3,1 , then the Liu-Yau mass of Σ is strictly positive unless Σ lies on a hyperplane. We also show that the examples given byÓ Murchadha, Szabados and Tod [18] are special cases of this result.