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Convergence of Riemannian manifolds with integral bounds on curvature. I. (English) Zbl 0748.53025
The main result of the paper is a generalization of the local Gromov convergence theorem. Here an integral L n/2 -bound for the curvature on geodesic balls with a lower volume bound replaces the pointwise bound. In the proof a lower bound for the isoperimetric constant for geodesic balls of a fixed radius is shown. Then the author introduces a local version of the Ricci flow, which allows to control local bounds for the curvature. Examples show, that it is necessary to assume local lower volume bounds in the main theorem.

MSC:
53C20Global Riemannian geometry, including pinching