Quantitative Volume Space From Rigidity with lower Ricci curvature bound II

This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is diffeomorphic to a $H$-space form if for every ball of definite size on $M$, the lifting ball on the Riemannian universal covering space of the ball achieves an almost maximal volume, provided the diameter of $M$ is bounded for $H\ne 1$. In [CRX], we verified the conjecture for the case that $M$ or its Riemannian universal covering space $\tilde M$ is not collapsed for $H=1$ or $H\ne 1$ respectively. In the present paper, we will verify this conjecture for the case that Ricci curvature is also bounded above, while the above non-collapsing condition is not required.