Volumes of balls in large Riemannian manifolds. (English) Zbl 1232.53032

The author proves two important lower bounds for the volumes of balls in a Riemannian manifold:
(1) For each dimension \(n\), there is a number \(\delta(n)>0\) such that the following estimate holds: If \((M^n,g)\) is a complete Riemannian manifold with filling radius at least \(R\), then it contains a ball of radius \(R\) and volume at least \(\delta(n)R^n\).
(2) For each dimension \(n\), there is a number \(\delta(n)>0\) such that the following estimate holds: If \((M^n,\text{hyp})\) is a closed hyperbolic manifold and if \(g\) is another metric on \(M\) with volume not greater than \(\delta(n)\text{Vol}(M,\text{hyp})\), then the universal cover of \((M,g)\) contains a unit ball with volume greater than the volume of a unit ball in hyperbolic \(n\)-space.


53C20 Global Riemannian geometry, including pinching
Full Text: DOI arXiv


[1] G. Besson, G. Courtois, and S. Gallot, ”Entropies et rigidités des espaces localement symétriques de courbure strictement négative,” Geom. Funct. Anal., vol. 5, iss. 5, pp. 731-799, 1995. · Zbl 0851.53032 · doi:10.1007/BF01897050
[2] E. Bombieri and L. Simon, ”On the Gehring link problem,” in Seminar on Minimal Submanifolds, Princeton, NJ: Princeton Univ. Press, 1983, vol. 103, pp. 271-274. · Zbl 0529.53005
[3] H. Federer and W. H. Fleming, ”Normal and integral currents,” Ann. of Math., vol. 72, pp. 458-520, 1960. · Zbl 0187.31301 · doi:10.2307/1970227
[4] M. Gromov, ”Large Riemannian manifolds,” in Curvature and Topology of Riemannian Manifolds, New York: Springer-Verlag, 1986, vol. 1201, pp. 108-121. · Zbl 0601.53038 · doi:10.1007/BFb0075649
[5] M. Gromov, ”Filling Riemannian manifolds,” J. Differential Geom., vol. 18, iss. 1, pp. 1-147, 1983. · Zbl 0515.53037
[6] M. Gromov, ”Systoles and intersystolic inequalities,” in Actes de la Table Ronde de Géométrie Différentielle, Paris: Soc. Math. France, 1996, vol. 1, pp. 291-362. · Zbl 0877.53002
[7] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Boston, MA: Birkhäuser, 1999, vol. 152. · Zbl 0953.53002
[8] M. Gromov, ”Volume and bounded cohomology,” Inst. Hautes Études Sci. Publ. Math., iss. 56, pp. 5-99 (1983), 1982. · Zbl 0516.53046
[9] M. Gromov, ”Positive curvature, macroscopic dimension, spectral gaps and higher signatures,” in Functional Analysis on the Eve of the 21st Century, Vol. II, Boston, MA: Birkhäuser, 1996, vol. 132, pp. 1-213. · Zbl 0945.53022
[10] M. Gromov and B. H. Lawson Jr., ”Positive scalar curvature and the Dirac operator on complete Riemannian manifolds,” Inst. Hautes Études Sci. Publ. Math., iss. 58, pp. 83-196 (1984), 1983. · Zbl 0538.53047 · doi:10.1007/BF02953774
[11] M. Katz, ”The filling radius of two-point homogeneous spaces,” J. Differential Geom., vol. 18, iss. 3, pp. 505-511, 1983. · Zbl 0529.53032
[12] M. Katz, ”The first diameter of \(3\)-manifolds of positive scalar curvature,” Proc. Amer. Math. Soc., vol. 104, iss. 2, pp. 591-595, 1988. · Zbl 0693.53013 · doi:10.2307/2047018
[13] C. LeBrun, ”Four-manifolds without Einstein metrics,” Math. Res. Lett., vol. 3, iss. 2, pp. 133-147, 1996. · Zbl 0856.53035 · doi:10.4310/MRL.1996.v3.n2.a1
[14] A. Malcev, ”On isomorphic matrix representations of infinite groups,” Rec. Math. [Mat. Sbornik], vol. 8 (50), pp. 405-422, 1940. · Zbl 0025.00804
[15] J. H. Michael and L. M. Simon, ”Sobolev and mean-value inequalities on generalized submanifolds of \(R^n\),” Comm. Pure Appl. Math., vol. 26, pp. 361-379, 1973. · Zbl 0256.53006 · doi:10.1002/cpa.3160260305
[16] R. M. Schoen, ”Variational theory for the total scalar curvature functional for Riemannian metrics and related topics,” in Topics in Calculus of Variations, New York: Springer-Verlag, 1989, vol. 1365, pp. 120-154. · Zbl 0702.49038 · doi:10.1007/BFb0089180
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.