On conformally compact Einstein manifolds.
During the last couple of years conformally compact Einstein manifolds have appeared in string theory as the mathematical framework for the Ads/CFT correspondence which gives a close connection between conformal field theory and supergravity. Inspired by these facts the author establishes several results which support an expectation that there should be a very interesting relationship between the geometry of a conformally compact Einstein manifold and the conformal geometry of its conformal infinity. Assuming \((M,g)\) is a conformally compact Riemannian manifold of dimension \(n+1\) with \(Ric\geq-n\) and \(\lambda_0(g)>n-1\), the author states \(H_n(M,{\mathbb Z})=0\). In particular, the conformal infinity is connected. But, if \(\lambda_0(g)=n-1\) and \(H_n(M,{\mathbb Z})\neq0\) then he proves \(M\) is isometric to \({\mathbb R}\times\Sigma\), with warped product metric \(dt^2+\cosh ^2(t)h\), where \(\Sigma\) is compact and \(h\) is a metric on \(\Sigma\) with \(Ric\geq-(n-1)\). The author also considers the following problem: Can a given conformal vector field on \(\Sigma\) be extended to a Killing vector field on \(M\)? Examples of conformally compact Einstein manifolds are given by convex cocompact hyperbolic manifolds, by a class of warped product \(B^k\times N\) of a ball and a Riemannian manifold of dimension \(n+1-k\) such that Ric\(\,=\,-(n-k)\), etc. The conformal infinity of conformally compact Einstein manifolds is studied too. A non-existence of a conformally compact Einstein metric with some special type of the conformal infinity is established.
scientific article; zbMATH DE number 1744225