Einstein structures: Existence versus uniqueness. (English) Zbl 0842.53032
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 [I. A. Volodin, V. E. Kuznetsov and 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.
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.
Reviewer: J.Girbau (Barcelona)
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 03F50 | Metamathematics of constructive systems |
Keywords:
existence of Einstein metrics; negative scalar curvature; homology sphere; fundamental groupReferences:
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