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

Abstract

We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown-York \cite{BYmass1} \cite{BYmass2} and Liu-Yau \cite{LY1} \cite{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 \cite{ADM61} 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 $\Phi (x)$ in the asymptotic region outside $S_r$. In the third part, we prove that for any closed, spacelike, 2-surface $\Sigma$ in the Minkowski space ${\mathbb{R}}^{3,1}$ for which the Liu-Yau mass is defined, if $\Sigma$ bounds a compact spacelike hypersurface in ${\mathbb{R}}^{3,1}$, then the Liu-Yau mass of $\Sigma$ is strictly positive unless $\Sigma$ lies on a hyperplane. We also show that the examples given by Ó Murchadha, Szabados and Tod \cite{MST} are special cases of this result.

28 pages

References in corpus (3)

• Isometric embeddings into the Minkowski space and new quasi-local mass
• On the volume functional of compact manifolds with boundary with constant scalar curvature
• The Liu-Yau mass as a good quasi-local energy in general relativity

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