On Geometric Problems Related to Brown-york and Liu-yau Quasilocal Mass

We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown-York [5] [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 Sr and an integral of the scalar curvature plus a geometrically constructed function Φ(x) in the asymptotic region outside Sr. In the third part, we prove that for any closed, spacelike, 2-surface Σ in the Minkowski space R for which the Liu-Yau mass is defined, if Σ bounds a compact spacelike hypersurface in R, then the Liu-Yau mass of Σ is strictly positive unless Σ lies on a hyperplane. We also show that the examples given by Ó Murchadha, Szabados and Tod [18] are special cases of this result.