A remark on the inequality of McKean
type: Item
instance of: scholarly article
title: A remark on the inequality of McKean
zbMATH Open document ID: 1060.53045
author: Alain R. Veeravalli
published in: Commentarii Mathematici Helvetici
publication date: 2004-02-05T00:00:00Z
review text: For a connected, simply connected, complete non-compact Riemannian manifold \((M,g)\) of dimension \(m\) let \(\lambda_1(M,g)\) be its first eigenvalue, i.e. the infimum by \(\delta\) among the spectral parameters \(\lambda\) for which the following Dirichlet problem is solvable: \(\Delta\phi+\lambda\phi=0\) on \(B(x,\delta)\), \(\phi=0\) on \(\partial B(x,\delta)\) (the choice of point \(x\in M\) is inessential in the limit). Let \(M={\mathbb R}\times N\), where \((N,g_0)\) is a complete Riemannian manifold of dimension \(m-1\). Equip \(M\) with the metric \(g=dt^2+e^{2w(t)}g_0\), where \(t\) is the parameter on \({\mathbb R}\). The first theorem states that if \(w'(t)\geq\kappa>0\), then \[ \lambda_1(M,g)\geq(m-1)^2\kappa^2/4. \] The result is similar to the inequality of McKean, who proved the above formula for open Riemannian manifolds of dimension \(m\) with sectional curvature \(\leq-\kappa^2<0\). Another theorem of the paper concerns bounds of \(\lambda_1(M',g| _{M'})\) for a submanifold \(M'\subset M\) via its mean curvature.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 2038414
zbMATH Keywords: Eigenvalues of the Laplacian
zbMATH Keywords: Dirichlet problem
reviewed by: Boris Kruglikov
MaRDI profile type: Publication
full work available at URL: s00014-003-0782-9
OpenAlex ID: W2293745035
DOI: 10.1007/S00014-003-0782-9
description: scientific article; zbMATH DE number 2038414