Volume and area renormalizations for conformally compact Einstein metrics

Proceedings of the 19th Winter School "Geometry and Physics", GDML_Books, (2000), p. 31-42 / Harvested from

Résumé

Let $X$ be the interior of a compact manifold $\bar{X}$ of dimension $n + 1$ with boundary $M = \partial X$ , and $g_{+}$ be a conformally compact metric on $X$ , namely $\bar{g} \equiv r^{2} g_{+}$ 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 $r = 0$ , $d r \neq 0$ on $M$ . The restrction of $\bar{g}$ to $T M$ 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 $(g_{+}) = - n g_{+}$ , which are called conformally compact Einstein metrics on $X$ , and their extensions to $X$ together with the restrictions of $\bar{g}$ 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

MR MR1758076 | Zbl 0984.53020

EUDML-ID : urn:eudml:doc:221675
Mots clés:

@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/