TY - JOUR

T1 - On Geometric Problems Related to Brown-York and Liu-Yau Quasilocal Mass

AU - Miao, Pengzi

AU - Shi, Yuguang

AU - Tam, Luen Fai

N1 - Funding Information:
Research partially supported by Grant of NSFC (10725101).
Funding Information:
Research partially supported by Australian Research Council Discovery Grant #DP0987650.
Funding Information:
Research partially supported by Hong Kong RGC General Research Fund #GRF 2160357.

PY - 2010/9

Y1 - 2010/9

N2 - 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 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 R3 for which the Liu-Yau mass is defined, if Σ bounds a compact spacelike hypersurface in R3,1 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 et al. (Phys Rev Lett 92:259001, 2004) are special cases of this result.

AB - 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 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 R3 for which the Liu-Yau mass is defined, if Σ bounds a compact spacelike hypersurface in R3,1 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 et al. (Phys Rev Lett 92:259001, 2004) are special cases of this result.

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U2 - 10.1007/s00220-010-1042-7

DO - 10.1007/s00220-010-1042-7

M3 - Article

AN - SCOPUS:77954622343

VL - 298

SP - 437

EP - 459

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -