L² 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.