On the Topology of Conformally Compact Einstein 4-manifolds

In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally compact Einstein 4-manifold is diffeomorphic to B and its conformal infinity is diffeomorphic to S. 0. Introduction Conformally compact Einstein manifolds play an important part in the AdS/CFT correspondence, a promising new area under extensive development in string theory [Mc] [GKP] [W]. Mathematically the study of conformally compact Einstein structures began with the work of Fefferman and Graham ([FG1]) in which they gave a procedure to find local conformal invariants and the associated conformally covariant operators. It has been a very challenging problem to give general existence theory. Recently M. Anderson ([A2]) has shown the existence of conformally compact Einstein metrics whose boundary is the 3-sphere with arbitrary conformal structure in the connected component of the round one, thus extending the perturbation existence result of Graham and Lee ([GL]). On the other hand, in recent works, using several conformally covariant differential equations and their ties to Chern-Gauss-Bonnet formula, two of the coauthors with Gursky studied Princeton University, Department of Mathematics, Princeton, NJ 08544-1000, supported by NSF Grant DMS–0070542. University of California, Department of Mathematics, Santa Cruz, CA 95064, supported in part by NSF Grant DMS–0103160. Princeton University, Department of Mathematics, Princeton, NJ 08544-1000, supported by NSF Grant DMS–0070526. Typeset by AMS-TEX 1 2 CONFORMALLY COMPACT EINSTEIN the conformal geometry of closed 4-manifolds and obtained remarkable progress in understanding of the topology of 4-manifolds. In our view, conformally compact Einstein 4-manifolds provide a platform to study conformal geometry of closed 3-manifolds. In this note we will at least use it to translate the results in four dimension to 3-manifolds. An oriented manifold (X, g) with boundary M is a conformally compact Einstein manifold if there is a complete Einstein metric g in the interior of X , and a smooth defining function s for the boundary M = ∂X = {s = 0} so that (X, sg) is a compact Riemannian manifold with boundary. Since defining functions are not unique the data (X, g) determines a conformal structure (M, [ĝ]) which is called the conformal infinity of (X, g). It will be advantageous to find a defining function s whose associated conformal metric enjoys optimal positivity property. To this end, one of us first observe ([Q]), Theorem. Suppose that (X, g) is a conformally compact Einstein manifold, and that ĝ is a Yamabe metric for (M, [ĝ]). Then there exists a conformal compactification (X, ug), which has a totally geodesic boundary M and whose scalar curvature R[ug] ≥ n+1 n−1R[ĝ]. An important invariant of the conformally compact Einstein structure is the renormalized volume (see section one for the precise definition). The volume of a conformally compact Einstein manifold is infinite. But an appropriate normalization gives rise to the new invariants as suggested in the works of Maldacena [Mc], Witten [W], and Gubser, Klebanov and Polyakov [GKP]. The renormalization was carried out by Henningson and Skenderis in [HS] and later also by Graham in [Gr]. Anderson in [A1] observed that the renormalized volume for a conformally compact Einstein 4-manifold appears in the Chern-Gauss-Bonnet formula (0.1) 8πχ(X) = 1 4 ∫ X |W|dvg + 6V, where W is the Weyl curvature and the norm of the Weyl tensor is given by |W|2 = WijklW; i.e., the usual definition when W is viewed as a section of ⊗4T M and V is the renormalized volume of a conformally compact Einstein 4-manifold (X, g). It follows that the renormalized volume is an invariant of the underlying conformal structure of the conformally compact Einstein manifold. On the other hand, we recall that for a closed 4-manifold (Y, g), we have the Chern-Gauss-Bonnet formula