Volume and Area Renormalizations for Conformally Compact Einstein Metrics
Volume and Area Renormalizations for Conformally Compact Einstein Metrics
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发表时间：
1999-09
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通讯作者：
Bin G Raham
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作者：
Bin G Raham
设X是n+1维紧致流形X的内部，g +$是X上的共形紧度量，即g +$是x上的连续扩张的（或具有某种程度的平滑性）作为$X$的度量，其中$r$表示$M$的定义函数，即$r>0$ on $X$和$r=0$，$dr\ne 0$ on $M$。$\overline g$到$TM$的重构在改变$r$时重新标度，因此不变地定义了$M$上的共形度量类，称为$g_+$的共形无穷大。本文考虑X上满足Einstein条件Ric$（g+）=-ng +$的共形紧度量，以及它们在X上的推广，并考虑g对边界M=\partial X$的限制.首先，作者指出，对于共形紧Einstein度量的共形无穷远，M上的一个代表性度量g
Let $X$ be the interior of a compact manifold $\overline X$ of dimension $n+1$ with boundary $M=\partial X$, and $g_+$ be a conformally compact metric on $X$, namely $\overline g\equiv r^2g_+$ extends continuously (or with some degree of smoothness) as a metric to $X$, where $r$ denotes a defining function for $M$, i.e. $r>0$ on $X$ and $r=0$, $dr\ne 0$ on $M$. The restrction of $\overline g$ to $TM$ rescales upon changing $r$, so defines invariantly a conformal class of metrics on $M$, which is called the conformal infinity of $g_+$. In the present paper, the author considers conformally compact metrics satisfying the Einstein condition Ric$(g_+)=-ng_+$, which are called conformally compact Einstein metrics on $X$, and their extensions to $X$ together with the restrictions of $\overline g$ to the boundary $M=\partial X$. First, the author notes that a representative metric $g$ on $M$ for the conformal infinity of a conformally compact Einstein metric