A note on eigenvalues
type: Item
instance of: scholarly article
title: A note on eigenvalues
zbMATH Open document ID: 0703.53038
author: Jiecheng Chen
author: Jiayu Li
published in: Chinese Science Bulletin
publication date: 1990-01-01T00:00:00Z
review text: Let M be a simply connected complete Riemannian manifold with sectional curvature \(K_ M\). Yau conjectured that if \(-k^ 2_ 2\leq K_ M\leq - k^ 2_ 1<0,\) then M has no \(L^ 2\) eigenvalues. The authors construct a counter example to this conjecture. Let \(0\leq k_ 1<k_ 2<\infty\). The authors construct a spherically symmetric metric \(ds^ 2=dr^ 2+f(r)^ 2d\theta^ 2\) on \(M=R^ n\) such that (1) \(-k^ 2_ 2\leq K_ M\leq -k^ 2_ 1\), (2) M has \(L^ 2\) eigenvalues.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4153383
zbMATH Keywords: sectional curvature
zbMATH Keywords: counter example
zbMATH Keywords: \(L^ 2\) eigenvalues
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4153383