An exact solution to an equation and the first eigenvalue of a compact manifold
type: Item
instance of: scholarly article
title: An exact solution to an equation and the first eigenvalue of a compact manifold
zbMATH Open document ID: 1147.58032
author: Jun Ling
published in: Illinois Journal of Mathematics
publication date: 2008-08-26T00:00:00Z
review text: Let \((M,g)\) be a closed Riemannian manifold of dimension \(n\) and let \(\lambda\) be the first non-zero eigenvalue of the scalar Laplacian. Assume that the Ricci curvature has a negative lower bound \((n-1)\kappa<0\). Let \(r\) be the largest interior radius of the nodal domains of the first eigenvalue. Let \(d=2r\). The author establishes the estimates: \(\lambda\geq{{\pi^2}\over{d^2[1-(n-1)\kappa/(2\lambda)]}}>0\) and \(\lambda\geq{\pi^2\over d^2}+{1\over2}(n-1)\kappa\,.\)
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 5316795
zbMATH Keywords: spectral geometry
zbMATH Keywords: negative Ricci curvature
zbMATH Keywords: nodal domains
zbMATH Keywords: eigenvalue estimate
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5316795