On Geometric Problems Related to Brown-York and Liu-Yau Quasilocal Mass

• DOI: 10.1007/s00220-010-1042-7
• Publication Date: 2010-04-09T03:29:48Z
• Authors: Pengzi Miao; Yuguang Shi; Luen-Fai Tam
• Keywords: Mathematics - Differential Geometry; 53C20; 58JXX; FOS: Physical sciences; General Relativity and Quantum Cosmology (gr-qc); 53C20; 58JXX; 01 natural sciences; General Relativity and Quantum Cosmology; Differential Geometry (math.DG); 0103 physical sciences; FOS: Mathematics; 0101 mathematics

28 pages<br/>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 $��(x)$ in the asymptotic region outside $S_r $. In the third part, we prove that for any closed, spacelike, 2-surface $��$ in the Minkowski space $\R^{3,1}$ for which the Liu-Yau mass is defined, if $��$ bounds a compact spacelike hypersurface in $\R^{3,1}$, then the Liu-Yau mass of $��$ is strictly positive unless $��$ lies on a hyperplane. We also show that the examples given by �� Murchadha, Szabados and Tod \cite{MST} are special cases of this result.<br/>