Einstein structures: Existence versus uniqueness
type: Item
instance of: scholarly article
title: Einstein structures: Existence versus uniqueness
zbMATH Open document ID: 0842.53032
DOI: 10.1007/BF01928216
author: Alexander Nabutovsky
published in: Geometric and Functional Analysis. GAFA
publication date: 1996-07-28T00:00:00Z
full work available at URL: https://eudml.org/doc/58184
review text: The paper is based on the following Novikov theorem: Let \(p: \mathbb{R}^n\to \mathbb{R}\) be a polynomial function with rational coefficients such that the zero set \(Z(p)\) of \(p\) is non-empty, compact and non-singular. For \(n\geq 5\) there is no algorithm deciding whether or not \(Z(p)\) is diffeomorphic to the sphere \(S^n\) (the degree and the vector of coefficients of \(p\) are regarded as the input data for the algorithm). A proof of this theorem is sketched in [\textit{I. A. Volodin}, \textit{V. E. Kuznetsov} and \textit{A. T. Fomenko}, Russ. Math. Surv. 29, 71-172 (1974); translation from Usp. Mat. Nauk 29, No. 5(179), 71-168 (1974; Zbl 0303.57002)]. A complete proof of this theorem is given in the appendix of the present article. The author uses Novikov's theorem to prove some results concerning the existence of Einstein metrics. The author introduces the following definition: a compact manifold \(M\) almost admits an Einstein metric of negative scalar curvature if there exists an infinite sequence of metrics \(\{\mu_i\}\) on \(M\) such that the \(C^0\)-norm of the tensor \(\text{Ric}(\mu_i)+ \mu_i\) tends to zero (note that if \(g'\) is an Einstein metric of negative scalar curvature, then multiplying this metric by a positive constant one gets an Einstein metric \(g\) of scalar curvature \(- n\), so that \(\text{Ric}(g)+ g= 0\)). With this definition the author states the following result: If for \(n\geq 5\) every \(n\)-dimensional smooth homology sphere with ``large'' fundamental group almost admits an Einstein metric of negative scalar curvature, then \(S^n\) almost admits an Einstein metric of negative scalar curvature (the existence of Einstein metrics of negative scalar curvature on \(S^n\) for \(n\geq 5\) is an open problem). He also obtains some other similar results.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 03F50
zbMATH DE Number: 750735
zbMATH Keywords: existence of Einstein metrics
zbMATH Keywords: negative scalar curvature
zbMATH Keywords: homology sphere
zbMATH Keywords: fundamental group
reviewed by: Joan Girbau
MaRDI profile type: Publication
cites work: Ricci Curvature Bounds and Einstein Metrics on Compact Manifolds
cites work: Algebraic models defined over \(\mathbb{Q}\) of differential manifolds
cites work: Q4720334
cites work: Finiteness Theorems for Riemannian Manifolds
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cites work: Negatively Ricci curved manifolds
cites work: Cheeger's finiteness theorem for diffeomorphism classes of Riemannian manifolds.
cites work: Q3480816
cites work: THE PROBLEM OF DISCRIMINATING ALGORITHMICALLY THE STANDARD THREE-DIMENSIONAL SPHERE
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description: scientific article; zbMATH DE number 750735