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Ricci curvature bounds and Einstein metrics on compact manifolds. (English) Zbl 0694.53045
Say that a space \(S\) of compact Riemannian manifolds is \(C^{1,\alpha}\) compact in the Lipschitz topology if every sequence \((M_ i)\) in \(S\) has a subsequence which converges, with respect to the Lipschitz distance, to a \(C^{\infty}\) manifold \(M\) with \(C^ 0\) Riemannian metric and \(C^{1,\alpha}\) distance function. The main result of this paper is as follows: If \(S\) is the space of all compact \(n\)-dimensional Riemannian manifolds \(M\) such that \(| \text{Ric}_ M| \leq c_ 1\) and such that 5 other geometric quantities, like volume and diameter, are bounded a priori by constants \(c_ i\), \(2\leq i\leq 6\), then \(S\) is \(C^{1,\alpha}\) compact in the Lipschitz topology.
As an application, the author obtains compactness results for moduli spaces of Einstein metrics. For example, the space of compact 4-dimensional Einstein manifolds \(M\) with Einstein constant 1 and such that \(\text{vol}_ M\geq c_ 1,\) shortest null-homotopic geodesic loop \(\geq c_ 2\) and \(b_ 2(M)\leq c_ 3\) is compact in the \(C^{\infty}\) topology.

MSC:
53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
58D17 Manifolds of metrics (especially Riemannian)
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