关于Brown-York和Liu-Yau拟局部质量的几何问题
On geometric problems related to Brown-York and Liu-Yau quasilocal mass
摘要：
本文讨论了与Brown-York \cite{BYmass1} \cite{BYmass2}和Liu-Yau \cite{LY1} \cite{LY2}提出的准局域质量定义有关的一些几何问题.我们的讨论包括三个部分。在第一部分中，我们提出了一个新的变分问题的紧流形上的边界，这是由布朗约克质量的研究。我们证明了这个变分问题的临界点正是静态度量。在第二部分中，我们导出了在三维流形中发展的光滑二维闭曲面族的Brown-York质量的导数公式。作为一个有趣的副产品，我们能够将渐近平坦3-流形的ADM质量\cite{ADM61}写成坐标球的布朗-约克质量$S_r$与标量曲率的积分加上$S_r $外渐近区域中的几何构造函数$Φ(x)$之和。在第三部分中，我们证明了对于定义了Liu-Yau质量的Minkowski空间$\R^{3,1}$中的任何闭的类空2-曲面$Σ$，如果$Σ$在$\R^{3,1}$中有一个紧的类空超曲面，则除非$Σ$位于超平面上，否则$Σ$的Liu-Yau质量是严格正的。我们还证明了Murchadha，Szabados和Tod \cite{MST}给出的例子是这个结果的特例.
英文摘要：
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 $Φ(x)$ in the asymptotic region outside $S_r $. In the third part, we prove that for any closed, spacelike, 2-surface $Σ$ in the Minkowski space $\R^{3,1}$ for which the Liu-Yau mass is defined, if $Σ$ bounds a compact spacelike hypersurface in $\R^{3,1}$, 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, Szabados and Tod \cite{MST} are special cases of this result.
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