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Quantitative volume space form rigidity under lower Ricci curvature bound. II. (English) Zbl 1394.53045
Let \(M\) be a compact \(n\)-manifold of Ricci curvature bounded below by \((n-1)H,\;H=\pm 1\) or \(0\). The authors investigate the following space form rigidity conjecture: \(M\) is diffeomorphic to an \(H\)-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 \(H\neq 1\).
This paper is the second part of an earlier paper [the authors, “Quantitative volume space form rigidity under lower Ricci curvature bound”, Preprint, arXiv:1604.06986]. In the first paper, the authors verified the conjecture for the case that the Riemannian universal covering space \(\tilde M\) is not collapsed. In the reviewed paper, they verify this conjecture for the case when Ricci curvature is also bounded above, but the non-collapsing condition on \(\tilde M\) is not required. The authors conclude this paper with three questions related to the approach in this paper.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C24 Rigidity results
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