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\(L^ p\) pinching and compactness theorems for compact Riemannian manifolds. (English) Zbl 0753.53027

It is studied how integral bounds on the curvature of a compact Riemannian manifold determine the topology. More precisely \(L^ p\)- bounds for the curvature with \(p>\dim(M)/2\) and a lower bound for the isoperimetric constant are assumed. The proof uses Hamilton’s Ricci flow and a global version of the Moser iteration. There are related results due to M. T. Anderson [Invent. Math. 102, No. 2, 429-445 (1990; Zbl 0711.53038)], L. Z. Gao [J. Differ. Geom. 32, No. 2, 349-381 (1990; Zbl 0752.53022); ibid., No. 1, 155-183 (1990; Zbl 0719.53024); ibid., No. 3, 713-774 (1990; Zbl 0721.53039)] and M. Min-Oo and E. Ruh [Comment. Math. Helv. 65, No. 1, 36-51 (1990; Zbl 0704.53031)].

MSC:

53C20 Global Riemannian geometry, including pinching
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