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Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature. (English) Zbl 0972.58009

Summary: It is well known that critical points of the total scalar curvature functional \({\mathcal S}\) on the space of all smooth Riemannian structures of volume 1 on a compact manifold \(M\) are exactly the Einstein metrics. When the domain of \({\mathcal S}\) is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also Einstein or isometric to a standard sphere.
In this paper we prove that \(n\)-dimensional critical points have vanishing \(n-1\) homology under a lower Ricci curvature bound for dimension less than 8.

MSC:

58E11 Critical metrics
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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