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Volume and area renormalizations for conformally compact Einstein metrics. (English) Zbl 0984.53020
Slovák, Jan (ed.) et al., The proceedings of the 19th Winter School “Geometry and physics”, Srní, Czech Republic, January 9-15, 1999. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 63, 31-42 (2000).
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 \(r=0\), \(dr\neq 0\) 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 \(g_+\) determines a special defining function \(r\) in a neighbourhood of \(M\) and an identification of a neighborhood of \(M\) in \(X\) with \(M\times [0,\varepsilon)\). In this identification, the metric \(\overline g\) takes the form \(\overline g=g_r+ dr^2\) on \(M\times [0,\varepsilon)\), for a 1-parameter family \(g_r\) of metrics on \(M\) and \(g_+=r^{-2} (g_r+dr^2)\). By making use of this formula, it follows that the volume element \(dv_{g+}\) is given by \[ dv_{g+}= r^{-n-1} (\det g_r/\det g)^{1/2} dv_gdr. \tag{*} \] Calculating the expansion of \((\det g_r/ \det g)^{1/2}\) and integrating (*) over \(M\) by using it, the author shows that as \(\varepsilon\to 0\), \(\text{Vol} (\{r >\varepsilon\})\) has an asymptotic expansion in negative powers of \(\varepsilon\), and a \(\log\varepsilon\) term if \(n\) is even, and further that the constant term, say \(V\), in the expansion (for \(n\) odd) and the integral of the coefficient of \(\log(1/ \varepsilon)\) term in the expansion over \(M\) (for \(n\) even) are both independent of the choice of representative conformal metric \(g\) on \(M\) (Theorem 3.1).
The constant \(V\) is called the renormalized volume of \(X\). If \(n\) is odd, it defines a conformal invariant on \(M\) as mentioned above. However, if \(n\) is even, \(V\) is not conformally invariant, therefore giving rise to a so-called conformal anomaly. The author gives the formulas for the constant \(L\) when \(n=4\), 6, and also calculates the renormalized volume \(V\) for \(\mathbb{H}^{n+1}\) when \(n\) is odd. In the last section, the author discusses the area renormalization for minimal submanifolds and obtains the analogue of the result (Theorem 3.1) in the volume case.
For the entire collection see [Zbl 0940.00040].

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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