On conformally compact Einstein manifolds.
type: Item
instance of: scholarly article
title: On conformally compact Einstein manifolds.
zbMATH Open document ID: 1053.53030
author: Xiaodong Wang
published in: Mathematical Research Letters
publication date: 2001-01-01T00:00:00Z
review text: 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.
reviewed by: Neda Bokan
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 83E30
Mathematics Subject Classification ID: 83E50
zbMATH DE Number: 1744225
zbMATH Keywords: Bochner technique
zbMATH Keywords: Yamabe invariant
zbMATH Keywords: sectional curvature
zbMATH Keywords: conformal infinity
zbMATH Keywords: homology vanishing theorem
MaRDI profile type: Publication
full work available at URL: mrl.2001.v8.n5.a9
OpenAlex ID: W2028178868
DOI: 10.4310/MRL.2001.V8.N5.A9
description: scientific article; zbMATH DE number 1744225