Mathematische Zeitschrift

, Volume 277, Issue 1–2, pp 293–304 | Cite as

On the spread of positively curved Alexandrov spaces

Article

Abstract

It was proved by F. Wilhelm that Gromov’s filling radius of closed positively curved manifolds with a uniform lower bound on sectional curvature attains the maximum with the round sphere. Recently the author proved that this is also the case for closed finite-dimensional Alexandrov spaces with a positive lower curvature bound. These were proved as a corollary of a comparison theorem for the invariant called spread of those spaces. In this paper, we extend the latter result to infinite-dimensional Alexandrov spaces.

Keywords

Alexandrov space Spread Filling radius Packing radius 

Mathematics Subject Classification (2000)

53C23 

References

  1. 1.
    Alexander, S., Kapovitch, V. Petrunin, A.: Alexandrov geometry. Forthcoming textbookGoogle Scholar
  2. 2.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI (2001)Google Scholar
  3. 3.
    Colding, T.: Large manifolds with positive Ricci curvature. Invent. Math. 124(1–3), 193–214 (1996)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. 1, 443–474 (1979)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983)MATHMathSciNetGoogle Scholar
  6. 6.
    Grove, K., Petersen, P.: A radius sphere theorem. Invent. Math. 112(3), 577–583 (1993)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Grove, K., Wilhelm, F.: Hard and soft packing radius theorems. Ann. Math. (2) 142(2), 213–237 (1995)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Kapovitch, V.: Perelman’s Stability Theorem. Surveys in Differential Geometry, vol. XI, pp. 103–136. International Press, Somerville, MA (2007)Google Scholar
  9. 9.
    Katz, M.: The filling radius of two-point homogeneous spaces. J. Differ. Geom. 18(3), 505–511 (1983)MATHGoogle Scholar
  10. 10.
    Mitsuishi, A.: A splitting theorem for infinite dimensional Alexandrov spaces with nonnegative curvature and its applications. Geom. Dedicata 144, 101–114 (2010)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Perelman, G.: Elements of Morse theory on Aleksandrov spaces. Algebra i Analiz 5(1), 232–241 (1993) (Russian); translation in St. Petersburg Math. J. 5(1), 205–213 (1994)Google Scholar
  12. 12.
    Plaut, C.: Metric Spaces of Curvature \(\ge k\). Handbook of Geometric Topology, pp. 818–898. North-Holland, Amsterdam (2002)Google Scholar
  13. 13.
    Takatsu, A., Yokota, T.: Cone structure of \(L^2\)-Wasserstein spaces. J. Topol. Anal. 4(2), 237–253 (2012)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Wilhelm, F.: On the filling radius of positively curved manifolds. Invent. Math. 107(3), 653–668 (1992)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Yamaguchi, T.: Simplicial volumes of Alexandrov spaces. Kyushu J. Math. 51(2), 273–296 (1997)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Yokota, T.: A rigidity theorem in Alexandrov spaces with lower curvature bound. Math. Ann. 353(2), 305–331 (2012)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Yokota, T.: On the filling radius of positively curved Alexandrov spaces. Math. Z. 273(1–2), 161–171 (2013)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyoto Japan
  2. 2.Mathematisches InstitutUniversity of MünsterMünsterGermany

Personalised recommendations