Complete manifolds with positive spectrum
type: Item
instance of: scholarly article
title: Complete manifolds with positive spectrum
zbMATH Open document ID: 1032.58016
author: Peter Li
author: Jiaping Wang
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: Let \(M\) be a complete Riemannian manifold of dimension \(n\geq 3\). Let \(\text{Ric}_{M}\) be the Ricci curvature of \(M\) and \(\lambda_{1}(M)\) the lower bound of the spectrum of the Laplacian on \(M\). Consider the following two conditions on \(M\): (1) \(\text{Ric}_{M}\geq -\frac{(n-1)}{n-2}\lambda_{1}(M)\), (2) \(\lambda_{1}(M)>0\). In the paper under review, the authors study the complete Riemannian manifolds that satisfy conditions (1) and (2). In particular, they show that if a complete Riemannian manifold of dimension \(n\geq 3\) satisfies (1) and (2), then either \(M\) has only one infinite volume end, or it is a warped product \(\mathbb R\times N\), where \(N\) is some compact manifold with Ricci curvature bounded from below by \(-(n-2)\). They also prove that if the curvature condition (1) is satisfied only at infinity with a strict inequality, then the manifold must have finitely many ends with infinite volume.
reviewed by: Salah Mehdi
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1782697
zbMATH Keywords: complete Riemannian manifold
zbMATH Keywords: Ricci curvature
zbMATH Keywords: splitting
zbMATH Keywords: end
Wikidata QID: Q115202217
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090348357
OpenAlex ID: W2162220340
DOI: 10.4310/JDG/1090348357
description: scientific article; zbMATH DE number 1782697