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Almost Einstein manifolds of negative Ricci curvature. (English) Zbl 0725.53050

By the technique used to deform the metric in the direction of its Ricci curvature, first done by R. S. Hamilton [Differ. Geom. 17, 255-306 (1982; Zbl 0504.53034)], the author proves the following pinching theorem for the Ricci curvature. Theorem: For any \(n\geq 3\) and \(\Lambda >0\), there exists an \(e(n,\Lambda)\), depending only on n and \(\Lambda\), such that if \((M^ n,g)\in M^-(m,\Lambda)\) and if its Ricci curvature satisfies \(\max_{SM} | n\rho /r-1| <\epsilon(n,\Lambda),\) then M admits an Einstein metric \(\bar g\) with \(\rho(\bar g)\equiv -1\), where \(M^-(n,\Lambda)\) denotes the set of all smooth compact Riemannian manifolds \((M^ n,g)\) satisfying) \(r<0\) and \(d^ 2 \max | K| \leq \Lambda^ 2\).

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Citations:

Zbl 0504.53034
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