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 rho > 0 such that for any x is an element of M, the open rho-ball at x* in the (local) Riemannian universal covering space, (U-rho*, x*) -> (B-rho(x), x), has the maximal volume, i.e., the volume of a rho-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].