On geometric problems related to Brown-York and Liu-Yau quasilocal mass

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We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown-York BYmass1 BYmass2 and Liu-Yau LY1 LY2. 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 ADM61 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 ³,¹ for which the Liu-Yau mass is defined, if Σ bounds a compact spacelike hypersurface in 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 by Ó Murchadha, Szabados and Tod MST are special cases of this result.