An eigenvalue bound for compact manifolds with varying curvature
type: Item
instance of: scholarly article
title: An eigenvalue bound for compact manifolds with varying curvature
zbMATH Open document ID: 0685.53036
DOI: 10.1007/BF00127861
published in: Annals of Global Analysis and Geometry
publication date: 1989-01-01T00:00:00Z
review text: Let \(\rho\) be the Ricci tensor of a compact m dimensional manifold M and let \(\lambda\) (x) be the lowest eigenvalue of \(\rho\) ; assume \(\lambda\) (x)\(\geq 0\) and \(\lambda\) \(\not\equiv 0\). Let \(\lambda_ 1\) be the lowest non-zero eigenvalue of the Laplacian \(\Delta =d\delta\). If \(\lambda\) (x)\(\geq \alpha\), Lichnerowicz showed Theorem 1 (Lichnerowicz): \(\lambda_ 1\geq m\alpha /(m-1).\) Let \(W(x)=2(\alpha -\alpha \lambda (x))\) where \(\alpha\) is constant and \(0\leq W\leq 2\alpha\). Let \(|_{p,q}\) be the norm of an operator from \(L^ p\) to \(L^ q\) and let \(c=| W(2\alpha +\Delta)^{- 1}|_{1,1};\quad 0\leq c<1.\) Theorem 2: The smallest non-zero eigenvalue \(\lambda_ 1\) of \(\Delta\) satisfies \(\lambda_ 1\geq m\alpha (1-c)/(m-1+c).\) Remark: If \(\rho\) (x)\(\geq \alpha\) for all \(x\in M\), then \(c=0\) so (2)\(\Rightarrow (1)\). Theorem 2 is a generalization of Lichnerowicz's result and gives a lower bound on the lowest eigenvalue if \(\rho\) (x)\(\geq \alpha\) on a large fraction of the manifold.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4122783
zbMATH Keywords: Ricci tensor
zbMATH Keywords: lowest non-zero eigenvalue
zbMATH Keywords: Laplacian
author: E. Brian Davies
MaRDI profile type: Publication
cites work: The splitting theorem for manifolds of nonnegative Ricci curvature
cites work: A lower bound for the heat kernel
cites work: Q4041031
cites work: Curvature, diameter and Betti numbers
cites work: Three-manifolds with positive Ricci curvature
cites work: Q5186326
cites work: Q3883196
cites work: On the parabolic kernel of the Schrödinger operator
cites work: Q5640160
full work available at URL: https://doi.org/10.1007/bf00127861
OpenAlex ID: W2043648779
description: scientific article; zbMATH DE number 4122783