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 $\overline{g}\equiv r^2{g}_{+}$ 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$\left(g_{+}\right)=-n{g}_{+}$, 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

EUDML-ID : urn:eudml:doc:221675
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@article{701645,
     title = {Volume and area renormalizations for conformally compact Einstein metrics},
     booktitle = {Proceedings of the 19th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {2000},
     pages = {31-42},
     mrnumber = {MR1758076},
     zbl = {0984.53020},
     url = {http://dml.mathdoc.fr/item/701645}
}

Graham, Robin C. Volume and area renormalizations for conformally compact Einstein metrics, dans Proceedings of the 19th Winter School "Geometry and Physics", GDML_Books,  (2000), pp. 31-42. http://gdmltest.u-ga.fr/item/701645/