Volume and area renormalizations for conformally compact Einstein metrics

summary: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 metri