On the stability of Riemannian manifold with parallel spinors
type: Item
instance of: scholarly article
title: On the stability of Riemannian manifold with parallel spinors
zbMATH Open document ID: 1075.53042
author: Xianzhe Dai
author: Xiaodong Wang
author: Guofang Wei
published in: Inventiones Mathematicae
publication date: 2005-08-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0311253
review text: The authors study the stability problem for the total scalar curvature functional restricted to the transversal directions, i.e., the space of conformal structures. They show that Riemannian manifolds with nonzero parallel spinors (which are necessarily Ricci flat) are stable when one relates this problem to the second variation of the total scalar curvature functional restricted to the traceless symmetric \(2\)-tensors. Theorem 1.1: If a compact Riemannian manifold \((M,g)\) has a cover which is spin and admits nonzero parallel spinors, then the Lichnerowicz Laplacian \(\mathbb{L}_g\) is positive semi-definite. Theorem 1.2: Let \((M,g)\) be a compact manifold which admits a spin cover with nonzero parallel spinors. Then \(g\) cannot be deformed to positive scalar curvature metrics. Theorem 1.3: Any scalar flat deformation of a Calabi-Yau metric on a compact manifold must be Calabi-Yau. In fact, any deformation with nonnegative scalar curvature of a Calabi-Yau metric on a compact manifold is necessarily a Calabi-Yau deformation. The same is true for the other special holonomy metrics (hyper-Kähler, \(G_2\), and Spin (7)). Theorem 1.4: The Calabi-Yau (and other special holonomy) metrics are local maxima for the Yamabe invariant.
Mathematics Subject Classification ID: 53C27
Mathematics Subject Classification ID: 58D17
Mathematics Subject Classification ID: 58E11
zbMATH DE Number: 2190622
zbMATH Keywords: parallel spinors
zbMATH Keywords: Ricci flat metrics
zbMATH Keywords: special holonomy metrics
zbMATH Keywords: stability of Riemannian metrics
zbMATH Keywords: Calabi-Yau metrics
reviewed by: Vasile Oproiu
MaRDI profile type: Publication
OpenAlex ID: W2159231972
arXiv ID: math/0311253
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Wikidata QID: Q115388533
DOI: 10.1007/S00222-004-0424-X
description: scientific article