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On geometric problems related to Brown-York and Liu-Yau quasilocal mass | | Pengzi Miao
; Yuguang Shi
; Luen-Fai Tam
; | | Date: |
30 Jun 2009 | | 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 $R^{3,1}$ for which the Liu-Yau mass is
defined, if $Sigma$ bounds a compact spacelike hypersurface in $R^{3,1}$,
then the Liu-Yau mass of $Sigma$ is strictly positive unless $Sigma$ lies on
a hyperplane. We also show that the examples given by ’{O} Murchadha, Szabados
and Tod cite{MST} are special cases of this result. | | Source: | arXiv, 0906.5451 | | Services: | Forum | Review | PDF | Favorites | |
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