Quantitative Volume Space Form Rigidity Under Lower Ricci Curvature Bound

Let $M$ be a compact $n$-manifold of ${\mathrm{Ric}}_M\ge (n-1)H$ ($H$ 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 $x\in M$, the open $\rho$-ball at $x^{\ast }$ in the (local) Riemannian universal covering space, $(U^_\rho,x^)\to (B_\rho(x),x)$,hasthemaximalvolumei\mathrm{.}e\mathrm{.},thevolumeofa$\rho$-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].