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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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