Quantitative Maximal Diameter Rigidity of Positive Ricci Curvature

In Riemannian geometry, the Cheng's maximal diameter rigidity theorem says that if a complete \(n\)-manifold \(M\) of Ricci curvature, \(\operatorname{Ric}_M\ge (n-1)\), has the maximal diameter \(\pi\), then \(M\) is isometric to the unit sphere \(S^n_1\). The main result in this paper is a quantitative maximal diameter rigidity: if \(M\) satisfies that \(\operatorname{Ric}_M\ge n-1\), \(\operatorname{diam}(M)\approx \pi\), and the Riemannian universal cover of every metric ball in \(M\) of a definite radius satisfies a Riefenberg condition, then \(M\) is diffeomorphic and bi-H\"older close to \(S^n_1\).