L2 curvature and volume renormalization of AHE metrics on 4-manifolds

Abstract

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 L2 norm of the Weyl curvature, on the space of such Einstein metrics.