Rigidity of manifolds with large volume (Q1319336)

The following sphere theorem is the main result of the paper: For every positive integers \(n, p > n/2\) and \(C > 0\) there exists a positive number \(\varepsilon = \varepsilon(n,p,C)\) such that the following holds: Every complete manifold \(M^ n\) whose Ricci curvature satisfies \(\text{Ric} \geq n - 1\) and \(\int_ M | \text{Ric}|^ p \leq C\) and whose volume satisfies \(\text{vol}(M) \geq (1 - \varepsilon) \text{vol}(S^ n)\) is \(C^ 1\)-diffeomorphic to the standard sphere. This extends a result by \textit{M. T. Anderson} [Invent. Math. 102, No. 2, 429-445 (1990; Zbl 0711.53038)] since the pointwise upper bound for the Ricci curvature is replaced by an \(L^ p\)-bound. The proof uses an estimate for the harmonic radius and a convergence result.