ar X iv : g r - qc / 0 20 40 87 v 1 2 9 A pr 2 00 2 Report on GR 16 , Session A 3 : Mathematical Studies of the Field Equations

In this report, which is an extended version of that appearing in the Proceedings of GR16, I will give a summary of the main topics covered in Session A.3. on mathematical relativity at GR16, Durban. The summary is mainly based on extended abstracts submitted by the speakers. I would like to thank all participants for their contributions and help with this summary. According to a conjecture by Maldacena[1], supported by a growing body of evidence, there is a correspondence between string theory in Anti–deSitter (AdS) spacetime and a conformal field theory (CFT) on the boundary of AdS. Stability considerations motivated by the AdS/CFT conjecture led to the conjecture that the boundary of asymptotically AdS space–times should be connected. The Riemannian version of this was proved by Witten and Yau[2] and generalized by Cai and Galloway[3]. The non–supersymmetric version of the AdS/CFT correspondence, leads to consideration of asymptotically locally AdS spacetimes with conformal boundary such that one spatial direction is compactified on a circle. If we allow nontrivial topology at infinity, the positive mass theorem for general relativity is not valid in general[4]. In case of negative cosmological constant, Λ < 0, a partial Cauchy surface in an asymptotically locally AdS n+1-dimensional space–time may have any orientable n − 1–manifold as boundary at infinity. The Kottler and Nariai metrics give explicit examples of static AdS metrics on manifolds with nontrivial topology at infinity[5]. These examples all have horizons. Horowitz and Myers[4] have found a static solution of the Einstein equations with negative cosmological constant, Λ < 0, which is asymptotically locally AdS