A Yamabe type problem on compact spin manifolds
type: Item
instance of: scholarly article
title: A Yamabe type problem on compact spin manifolds
zbMATH Open document ID: 1053.58011
author: Bernd Ammann
author: Emmanuel Humbert
author: Bertrand Morel
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-08-20T00:00:00Z
review text: Let \((M,g,\sigma)\) be a compact spin manifold of dimension \(n\geq 2\). For a metric \(\tilde{g}\) in the conformal class \([g]\) of \(g\), let \(\lambda_1^+(\tilde{g})\) [resp. \(\lambda_1^-(\tilde{g})\)] be the smallest positive (resp. largest negative) eigenvalue of the Dirac operator \(D\) in the metric \(\tilde{g}\). The conformal invariants \(\lambda_{\min}^+(M,[g],\sigma):=\inf_{\tilde{g}\in [g]}\lambda_1^+(\tilde{g}) \text{Vol}(M,\tilde{g})^{1/n}\) and \(\lambda_{\min}^-(M,[g],\sigma):=\inf_{\tilde{g}\in [g]}| \lambda_1^-(g)|\text{Vol}(M,\tilde{g})^{1/n}\) have been studied by many authors [for example, see \textit{O. Hijazi}, Commun. Math. Phys. 104, 151--162 (1986; Zbl 0593.58040); \textit{J. Lott}, Pac. J. Math. 125, 117--126 (1986; Zbl 0605.58044); \textit{C. Bär}, Math. Ann. 293, 39--46 (1992; Zbl 0741.58046); the first author, ``The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions'' (Preprint, arXiv math. DG/0309061) and ``A variational problem in conformal spin geometry'' (Habilitationsschrift, Univ. Hamburg, May 2003)]. The authors show that the inequalities \[ \lambda_{\min}^+(M,[g],\sigma)\leq \lambda_{\min}^+(S^n)=\frac{n}{2}\omega_n^{1/n}\tag{1} \] and \[ \lambda_{\min}^-(M,[g],\sigma)\leq \lambda_{\min}^-(S^n)=\lambda_{\min}^+(S^n)\tag{2} \] are true, where \(\omega_n\) denotes the volume of the unit standard sphere \(S^n\) (see also the first author, loc. cit.). Moreover, if \((M,g)\) is nonconformally flat and \(n\geq 7\), then both inequalities are strict. If \((M,g)\) is conformally flat, \(D\) is invertible and the mass endomorphism [for this concept, see also the authors, ``Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds'' (Preprint Inst. É. Cartan, Nancy 2003/58)] possesses a negative (resp. positive) eigenvalue, then inequality (1) [resp. (2)] is also strict.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C27
zbMATH DE Number: 2093999
zbMATH Keywords: compact spin manifold
zbMATH Keywords: conformal class of metrics
zbMATH Keywords: Dirac operator
zbMATH Keywords: eigenvalue
zbMATH Keywords: mass endomorphism
zbMATH Keywords: conformal invariant
reviewed by: Mircea Craioveanu
MaRDI profile type: Publication
full work available at URL: j.crma.2004.03.018
OpenAlex ID: W1998855638
cites work: A spin-conformal lower bound of the first positive Dirac eigenvalue
cites work: The smallest Dirac eigenvalue in a spin-conformal class and cmc immersions
cites work: Positive mass theorem for the Yamabe problem on spin manifolds
cites work: Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds
cites work: Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire
cites work: Lower eigenvalue estimates for Dirac operators
cites work: Spineurs, opérateurs de Dirac et variations de métriques. (Spinors, Dirac operators and variations of the metrics)
cites work: Q4220320
cites work: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors
cites work: Q3985740
cites work: The Yamabe problem
cites work: Eigenvalue bounds for the Dirac operator
cites work: Critical metrics for the determinant of the Laplacian in odd dimensions
cites work: Conformal deformation of a Riemannian metric to constant scalar curvature
DOI: 10.1016/J.CRMA.2004.03.018
Recommended article: Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds
Recommended article: A spinorial analogue of Aubin's inequality
Recommended article: On a spin conformal invariant on manifolds with boundary
Recommended article: Q4719829
Recommended article: Q2969988
Recommended article: A spin-conformal lower bound of the first positive Dirac eigenvalue
Recommended article: Almost harmonic spinors
Recommended article: The spinorial τ-invariant and 0-dimensional surgery
Recommended article: Immersions minimales, première valeur propre du laplacien et volume conforme. (Minimal immersions, first eigenvalue of the Laplacian and conformal volume)
Recommended article: The smallest Dirac eigenvalue in a spin-conformal class and cmc immersions
description: scientific article; zbMATH DE number 2093999