L$^{\bf 2}$ Curvature and Volume Renormalization of AHE Metrics on 4-manifolds

This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds $M$ with the renormalized volume $V$, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on $M$. In addition we compute and discuss the differential or variation $dV$ of $V$, or equivalently the variation of the $L^{2}$ norm of the Weyl curvature, on the space of such Einstein metrics.

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Published 1 January 2001