On the Topology of Conformally Compact Einstein 4-manifolds

نویسندگان

  • Sun-Yung A. Chang
  • Paul Yang
چکیده

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

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Warped product and quasi-Einstein metrics

Warped products provide a rich class of physically significant geometric objects. Warped product construction is an important method to produce a new metric with a base manifold and a fibre. We construct compact base manifolds with a positive scalar curvature which do not admit any non-trivial quasi-Einstein warped product, and non compact complete base manifolds which do not admit any non-triv...

متن کامل

Harmonic Maps and the Topology of Conformally Compact Einstein Manifolds

We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.

متن کامل

Einstein Metrics with Prescribed Conformal Infinity on 4-manifolds

This paper considers the existence of conformally compact Einstein metrics on 4manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability depends on the topology of the filling manifold. The obstruction to extending the...

متن کامل

Topics in Conformally Compact Einstein Metrics

Conformal compactifications of Einstein metrics were introduced by Penrose [38], as a means to study the behavior of gravitational fields at infinity, i.e. the asymptotic behavior of solutions to the vacuum Einstein equations at null infinity. This has remained a very active area of research, cf. [27], [19] for recent surveys. In the context of Riemannian metrics, the modern study of conformall...

متن کامل

Construction of conformally compact Einstein manifolds

We produce some explicit examples of conformally compact Einstein manifolds, whose conformal compactifications are foliated by Riemannian products of a closed Einstein manifold with the total space of a principal circle bundle over products of Kähler-Einstein manifolds. We compute the associated conformal invariants, i.e., the renormalized volume in even dimensions and the conformal anomaly in ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2008