Quantitative volume space form rigidity under lower Ricci curvature bound I

Let M be a compact n-manifold of Ric_M ≥ (n − 1)H (H is a constant). We are concerned with the following space form rigidity: M is isometric to a space form of constant curvature H under either of the following conditions: (i) There is ρ > 0 such that for any x ∈ M, the open ρ-ball at x^∗ in the (local) Riemannian universal covering space, (U_ρ ^∗, x^∗) → (B_ρ(x), x), has the maximal volume, i.e., the volume of a ρ-ball in the simply connected n-space form of curvature H. (ii) For H = −1, the volume entropy of M is maximal, i.e., n − 1 ([LW1]). The main results of this paper are quantitative space form rigidity, i.e., statements that M is diffeomorphic and close in the Gromov–Hausdorff topology to a space form of constant curvature H, if M almost satisfies, under some additional condition, the above maximal volume condition. For H = 1, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].