An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds
type: Item
instance of: scholarly article
title: An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds
zbMATH Open document ID: 0722.53036
DOI: 10.1007/BF02571352
published in: Mathematische Zeitschrift
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174234
review text: Let \((M^ n,g)\) be a compact Riemannian spin manifold. The author proves several upper bounds depending on the sectional curvature and the injectivity radius for the first eigenvalue of the Dirac operator. These results are obtained by a comparison argument with the sphere. Let \(0=\lambda^ 2_ 0<\lambda^ 2_ 1<\lambda^ 2_ 2..\) denote the eigenvalues of \(D^ 2\) and let \(m_ j\) be the multiplicity of \(\lambda^ 2_ j\). Consider a smooth map \(f: M^{2m}\to S^{2m}\) with degree \(\deg (f)\geq 2^{m-1}(m_ 0+...+m_{k-1})+1.\) Then the k-th eigenvalue of the Dirac operator is bounded by \(| \lambda_ k| \leq 2^{m-1}\sqrt{m}\| df\|_{\infty}\) where \(\| df\|_{\infty}=\max | df_ x|.\)
reviewed by: Thomas Friedrich
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4190615
zbMATH Keywords: spin manifold
zbMATH Keywords: upper bounds
zbMATH Keywords: sectional curvature
zbMATH Keywords: injectivity radius
zbMATH Keywords: first eigenvalue
zbMATH Keywords: Dirac operator
author: Helga Baum
MaRDI profile type: Publication
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OpenAlex ID: W2081761147
description: scientific article; zbMATH DE number 4190615