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Quantitative maximal local rewinding volume rigidity with Ricci curvature bounded below II
ÕªÒª: For a metric ball $B_r(x)$ in a Riemannian manifold, its local rewinding volume is the volume of $B_r(x^*)$, where $B_r(x^*)\subset (U^*, x^*)$, the (uncomplete) Riemannian universal cover of $(B_r(x), x)$. A compact manifold with Ricci curvature bounded below by $(n-1)H$ is isometric to a space form with constant curvature $H$ if and only if every $\rho$-ball ($\rho$ fixed) achieves the maximal local rewinding volume. In this talk, we will prove that if a compact manifold $M$ with Ricci curvature lower bound $(n-1)H$ satisfies that the universal cover space $\tilde M$ is non-collapsing (there exist a positive lower bound of the volume of a unit ball in $\tilde M$) and each $\rho$-ball almost achieves the maximal local rewinding volume, then this manifold is diffeomorphic and close to a space form with $H$-constant curvature.