Smooth diameter and eigenvalue rigidity in positive Ricci curvature

A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given and there exists a positive constant such that any -dimensional complete Riemannian manifold with Ricci curvature , sectional curvature and diameter is Lipschitz close and diffeomorphic to the standard unit -sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.