On geometric problems related to brown-york and liu-yau quasilocal mass

Pengzi Miao, Yuguang Shi, Luen-Fai Tam

Research output: Contribution to journalArticleResearchpeer-review

15 Citations (Scopus)


We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown and York (Contemporary mathematics, vol 132, American Mathematical Society, Providence, pp 129-142, 1992; Phys Rev D (3) 47(4): 1407-1419, 1993) and Liu and Yau (Phys Rev Lett 90(23): 231102, 2003; J Am Math Soc 19(1): 181-204, 2006). 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 two dimensional surfaces evolving in an ambient three dimensional manifold. As a by-product, we are able to write the ADM mass (Arnowitt et al. in Phys. Rev. (2), 122: 997-1006, 1961) 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 Sr. In the third part, we prove that for any closed, spacelike, 2-surface Sigma in the Minkowski space R-3,R-1 for which the Liu-Yau mass is defined, if Sigma bounds a compact spacelike hypersurface in R-3,R-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 O Murchadha et al. (Phys Rev Lett 92: 259001, 2004) are special cases of this result.
Original languageEnglish
Pages (from-to)437 - 459
Number of pages23
JournalCommunications in Mathematical Physics
Issue number2
Publication statusPublished - 2010

Cite this