Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature
type: Item
instance of: scholarly article
title: Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature
zbMATH Open document ID: 1016.58016
published in: Pacific Journal of Mathematics
publication date: 2000-06-22T00:00:00Z
review text: Let \(M\) be a compact \(n\) dimensional Riemannian manifold with Ricci tensor \(\rho\) and diameter \(d\). The boundary, if any, is said to be weakly convex if the second fundamental form with respect to the outward normal is nonnegative; let \(\kappa\) be the associated mean curvature and let \(r\) be the inscribed radius. Impose suitable boundary conditions and let \(\lambda\) be the first positive eigenvalue of the scalar Laplacian. \textit{A. Lichnérowicz} [`Géométrie des groupes de transformations' (1984; Zbl 0096.16001)] showed that if \(M\) is closed and if \(\rho\geq(n-1)K>0\), then \(\lambda\geq nK\). The author generalizes work of \textit{P. Li} and \textit{S.-T. Yau} [Proc. Symp. Pure Math. 36, 205-239 (1980; Zbl 0441.58014)] and also of \textit{H. Yang} and \textit{J. Zhong} [Sci. Sin., Ser. A 27, 1265-1273 (1984; Zbl 0561.53046)] to establish two related results: Theorem 1. If \(M\) is closed and if \(\rho\geq(n-1)K\geq 0\), then \(\lambda\geq{1\over 4}(n-1)K+{{\pi^2}\over{d^2}}\). Theorem 2. Assume that \(\partial M\) is non-empty and that \(\rho\geq(n-1)K\geq 0\). a) If \(\partial M\) is weakly convex, then the first positive Neumann eigenvalue \(\lambda\) of \(M\) satisfies \(\lambda\geq{1\over 4}(n-1)K+{{\pi^2}\over{d^2}}\). b) If \(\kappa\geq 0\), then the first positive Dirichlet eigenvalue \(\lambda\) satisfies \(\lambda\geq{1\over 4}\{(n-1)K+{{\pi^2}\over {r^2}}\}\).
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58C40
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1463708
zbMATH Keywords: Dirichlet Laplacian
zbMATH Keywords: Neumann Laplacian
zbMATH Keywords: weakly convex boundary
zbMATH Keywords: non-negative mean curvature
zbMATH Keywords: gradient estimate technique
zbMATH Keywords: lower eigenvalue bound
zbMATH Keywords: non-negative Ricci curvature
author: Da-Gang Yang
MaRDI profile type: Publication
full work available at URL: pjm.1999.190.383
OpenAlex ID: W2080066812
DOI: 10.2140/PJM.1999.190.383
description: scientific article; zbMATH DE number 1463708