2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60303This 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.15 pages, several corrections made. To appear in Math. Research LettersDifferential Geometrytext