L 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 norm of the Weyl curvature, on the space of such Einstein metrics. 0. Introduction. The Chern-Gauss-Bonnet formula for a compact Riemannian 4-manifold (M,g) without boundary states that 1 8π2 ∫ M (|R| − 4|z|)dV = 1 8π2 ∫ M (|W | − 1 2 |z| + 1 24 s)dV = χ(M), (0.1) where R,W, z, s are respectively the Riemann, Weyl, trace-free Ricci and scalar curvatures. In particular, if g is an Einstein metric, then z = 0, and so Einstein metrics minimize the L2 norm of the curvature over all metrics on M . Hence the L2 norm of the full curvature of an Einstein metric on M is apriori bounded by the topology of M . If (M,g) is a compact 4-manifold with non-empty boundary, then (0.1) no longer holds; there is a correction or defect term given by certain curvature integrals over the boundary ∂M. If (M,g) is complete and open, then the boundary integrals relate to the asymptotic geometry of (M,g). When (M,g) is a complete non-compact, Ricci-flat 4-manifold, then the defect term in (0.1) is easily identified if the manifold asymptotically approaches that of a quotient of R4, i.e. M is asymptotically locally Euclidean (ALE), flat, (AF) or locally flat, (ALF), c.f. [5]. In this paper, we consider this issue when (M,g) is an Einstein 4-manifold of negative scalar curvature, which is asymptotically hyperbolic. To define this, let M be an arbitrary compact, connected and oriented 4-manifold with non-empty boundary ∂M ; we do not assume that ∂M is connected. According to Penrose, c.f. [14] and also [12], a complete metric g on M is conformally compact if there is a smooth defining function ρ on M̄ = M ∪ ∂M, i.e. ρ(∂M) = 0, dρ 6= 0 on ∂M and ρ > 0 on M , such that the metric ḡ = ρ · g, (0.2) extends to a smooth metric on M̄. We require that ḡ is at least C3 smooth up to ∂M, although this condition could be relaxed somewhat. Conversely, if ḡ is any smooth Riemannian metric on M̄ and ρ is any C1 defining function, then g ≡ ρ−2 · ḡ gives a complete conformally compact metric on the open manifold M . The defining function ρ is not unique, since it can be multiplied by any smooth positive function on M̄. Hence, both the metric ḡ and its induced metric γ on ∂M are not uniquely defined by (M,g). However, the conformal class [γ] of the metric γ = ḡT∂M is uniquely determined by the complete Partially supported by NSF Grant DMS 0072591. 2000 Math. Sci. Classification. Primary: 53C25, 53C80, Secondary: 58J60. 1