## 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

### Keywords:

comparison theorem; Gromov-Hausdorff distance

Zbl 0959.53013
Full Text:

### References:

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