Quantitative maximal diameter rigidity of positive Ricci curvature (Q6582266)

Let \(M\) be a complete Riemannian manifold of dimension \(n \geq 2\) whose Ricci curvature is bounded from below by \(n-1\). According to \textit{S.-Y. Cheng}'s Maximal Diameter Theorem [Math. Z. 143, 289--297 (1975; Zbl 0329.53035)], the diameter of \(M\) is less than or equal to \(\pi\), with equality holding if and only if \(M\) is isometric to the unit \(n\)-sphere. The main result of the paper under review is to prove that if the diameter of \(M\) is sufficiently close to \(\pi\), and if \(M\) satisfies a certain local regularity condition, that is, every point of \(M\) is a local rewinding Reifenberg point, then \(M\) is bi-Hölder diffeomorphic to the unit \(n\)-sphere.