A new proof of Lee's theorem on the spectrum of conformally compact Einstein manifolds
type: Item
instance of: scholarly article
title: A new proof of Lee's theorem on the spectrum of conformally compact Einstein manifolds
zbMATH Open document ID: 1025.58011
author: Xiaodong Wang
published in: Communications in Analysis and Geometry
publication date: 2003-04-07T00:00:00Z
review text: Let \(M\) be a compact \((m+1)\)-dimensional manifold with boundary \(\Sigma\). One says a Riemannian metric on \(\text{int}(M)\) is conformally compact if for any defining function \(r\) of the boundary, \(\tilde g:=r^2g\) extends to a \(C^3\) metric on all of \(M\). The conformal class of \(\tilde g _\Sigma\) is invariantly defined and is called the conformal infinity of \(g\). Mazzeo showed the spectrum of such a manifold is the ray \([m^2/4,\infty)\) with no embedded eigenvalues -- there can be finitely many eigenvalues in the interval \((0,n^2/4)\). \textit{J. Lee} [Commun. Anal. Geom. 3, 253-271 (1995; Zbl 0934.58029)] showed that if the conformal infinity has non-negative Yamabi invariant, then \(\lambda_0(g)=n^2/4\). In the present paper, the author uses ideas of Witten-Yau and Cai-Galloway to give a simple proof of this result.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1893859
MaRDI profile type: Publication
full work available at URL: cag.2002.v10.n3.a7
OpenAlex ID: W2621362919
DOI: 10.4310/CAG.2002.V10.N3.A7
description: scientific article; zbMATH DE number 1893859