Quantitative maximal diameter rigidity of positive Ricci curvature

Abstract

Abstract In Riemannian geometry, the Cheng’s maximal diameter rigidity theorem says that if a complete n-manifold M of Ricci curvature, Ric M ≥ ( n - 1 ) {\operatorname{Ric}_{M}\geq(n-1)} , has the maximal diameter π, then M is isometric to the unit sphere S 1 n {S^{n}_{1}} . The main result in this paper is a quantitative maximal diameter rigidity: if M satisfies that Ric M ≥ n - 1 {\operatorname{Ric}_{M}\geq n-1} , diam ⁡ ( M ) ≈ π {\operatorname{diam}(M)\approx\pi} , and the Riemannian universal cover of every metric ball in M of a definite radius satisfies a Reifenberg condition, then M is diffeomorphic and bi-Hölder close to S 1 n {S^{n}_{1}} .