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Manifolds with minimal radial curvature bounded from below and big volume. (English) Zbl 0970.53023

This paper deals with compact Riemannian manifolds \(M^n\) with radial curvature \(\geq 1\) (with some base point), as a general assumption. In a previous paper by the author and S. Mendonca [Indiana Univ. Math. J. 48, 249-274 (1999; Zbl 0959.53013)] it was shown that for sufficiently small \(\varepsilon>0\) the additional assumption of a lower bound \(\pi-\varepsilon\) for the radius implies that \(M\) is homeomorphic to the sphere \(S^n\) and that the Gromov-Hausdorff distance between \(M^n\) and \(S^n\) tends to 0 as \(\varepsilon\) tends to 0. In the present paper the following is shown: If the Gromov-Hausdorff distance between \(M^n\) and \(S^n\) is smaller than \(\varepsilon\) then the volume of \(M^n\) converges to the volume of \(S^n\) as \(\varepsilon\) converges to 0. Furthermore under the additional assumption \(\text{vol}(M^n) \geq\text{vol} (S^n)-\varepsilon\) the Gromov-Hausdorff distance between \(M^n\) and \(S^n\) tends to 0 as \(\varepsilon\) tends to 0. As a corollary, under the general assumption above, a convergence in the Gromov-Hausdorff distance turns out to be equivalent to a volume convergence.

MSC:

53C20 Global Riemannian geometry, including pinching
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Citations:

Zbl 0959.53013
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References:

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