Quantitative Maximal Diameter Rigidity of Positive Ricci Curvature

TL;DR

This paper establishes a quantitative version of Cheng's maximal diameter rigidity theorem for manifolds with Ricci curvature at least (n-1), showing that near-maximal diameter and certain local conditions imply the manifold is close to a sphere.

Contribution

It introduces a quantitative maximal diameter rigidity result, linking near-maximal diameter, Ricci curvature bounds, and local Riefenberg conditions to the manifold's topological and geometric closeness to a sphere.

Findings

Manifolds with Ricci curvature ≥ n-1 and diameter close to π are topologically similar to spheres.

Local Riefenberg conditions on metric balls imply global geometric rigidity.

The manifold is diffeomorphic and bi-Hölder close to the unit sphere.

Abstract

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 $S^n_1$.