# Article

 Title: Volume and area renormalizations for conformally compact Einstein metrics (English) Author: Graham, Robin C. Language: English Journal: Proceedings of the 19th Winter School "Geometry and Physics" Volume: Issue: 1999 Year: Pages: 31-42 . Category: math . 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 metric (English) MSC: 53C25 MSC: 53C80 MSC: 81T30 MSC: 81T40 MSC: 81T50 idZBL: Zbl 0984.53020 idMR: MR1758076 . Date available: 2009-07-13T21:42:40Z Last updated: 2012-09-18 Stable URL: http://hdl.handle.net/10338.dmlcz/701645 .

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