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Convergence and rigidity of manifolds under Ricci curvature bounds. (English) Zbl 0711.53038
Convergence and rigidity results for manifolds with bounded Ricci curvature and either bounded injectivity radius or volume are proved. It is shown that the space of compact Riemannian n-dimensional manifolds with bounded Ricci curvature, bounded diameter and a lower bound for the injectivity radius is compact in the $$C^{1,\alpha}$$-topology. This generalizes the Cheeger-Gromov compactness theorem. As an application a sphere theorem for manifolds with positive Ricci curvature and volume nearby the volume of the standard sphere is proved. This generalizes previous results since no bound on the sectional curvature is used. Using the compactness result it is also shown that a compact 4-dimensional almost Einstein manifold with positive Ricci curvature is close to an Einstein orbifold in the Gromov-Hausdorff topology.
The following observation is used in the proofs: The bounds for the Ricci curvature and the injectivity radius imply a lower bound on the size of balls with harmonic coordinates with $$C^{1,\alpha}$$-bounds on the metric.