References
Aleksandrov, A. D.,Die innere Geometrie der konvexen Flächen. Akademie-Verlag, Berlin, 1955.
Burago, Y., Gromov, M. & Perelman, G., A. D. Alexandrov's spaces with curvatures bounded from below, I. Preprint.
Calabi, E. & Cao, J., Simple closed geodesics on convex surfaces. Preprint.
Chapman, T. A. &Ferry, S., Approximating homotopy equivalences by homeomorphisms.Amer. J. Math., 101 (1979), 583–607.
Durumeric, O., Manifolds of almost half of the maximal volume.Proc. Amer. Math. Soc., 104 (1988), 277–283.
Ferry, S., Homotoping ε-maps to homeomorphisms.Amer. J. Math., 101 (1979), 567–582.
Gromov, M., Groups of polynomial growth and expanding maps. 53 (1981), 53–73.
— Filling Riemannian manifolds.J. Differential Geom., 18 (1983), 1–148.
Grove, K. &Petersen, V. P., Bounding homotopy types by geometry.Ann. of Math., 128 (1988), 195–206.
— Homotopy types of positively curved manifolds with large volume.Amer. J. Math., 110 (1988), 1183–1188.
— Manifolds near the boundary of existence.J. Differential Geom., 33 (1991), 379–394.
Grove, K. & Petersen V. P. On the excess of metric spaces and manifolds. Preprint.
— A pinching theorem for homotopy spheres.J. Amer. Math. Soc., 3 (1990), 671–677.
Grove, K., Petersen, V. P. &Wu, J.-Y., Geometric finiteness theorems via controlled topology.Invent. Math., 99 (1990), 205–213.
Grove, K. &Shiohama, K., A generalized sphere theorem.Ann. of Math., 106 (1977), 201–211.
Livesay, G. R., Fixed point free involutions on the 3-sphere.Ann. of. Math., 72 (1960), 603–611.
Otsu, Y., Shiohama, K. &Yamaguchi, T., A new version of differentiable sphere theorem.Invent. Math., 98 (1989), 219–228.
Petersen, V. P., A finiteness theorem for metric spaces.J. Differential Geom., 31 (1990), 387–395.
Quinn, F., Ends of maps, I.Ann. of Math., 110 (1979), 275–331.
—, Ends of maps, III: Dimensions 4 and 5.J. Differential Geom., 17 (1982), 503–521.
Rinow, W.,Die innere Geometrie der metrischen Räume. Springer-Verlag, Berlin, 1961.
Sakai, T., On the isodiametric inequality for the 2-sphere, inGeometry of Manifolds (ed. K. Shiohama). Perspectives in Math., 8. Academic Press, 1989, pp. 303–315.
Shioya, T., Diameter and area estimates forS 2 andP 2 with nonnegatively curved metrics. Preprint.
Shiohama, K. & Yamaguchi, T., Positively curved manifolds with restricted diameters, inGeometry of Manifolds (ed. K. Shiohama). Perspectives in Math., 8. Academic Press, 1989, pp. 345–350.
Wu, J.-Y., A volume/diameter-ratio for positively curved manifolds.Michigan Math. J., 37 (1990), 235–239.
Yamaguchi, T., Lipschitz convergence of manifolds of positive Ricci curvature with large volume.Math. Ann., 284 (1989), 423–436.
—, Collapsing and pinching under a lower curvature bound.Ann. of Math., 133 (1991), 317–357.
Yau, S. T.,Seminar on Differential Geometry. Ann. of Math. Stud., 102, Princeton, 1982.
Author information
Authors and Affiliations
Additional information
Supported in part by a grant from the National Science Foundation.
Supported in part by a grant from the National Science Foundation and the Sloan Foundation.
Rights and permissions
About this article
Cite this article
Grove, K., Petersen, P. Volume comparison à la Aleksandrov. Acta Math. 169, 131–151 (1992). https://doi.org/10.1007/BF02392759
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02392759