On the spread of positively curved Alexandrov spaces Authors Takumi Yokota Research Institute for Mathematical Sciences Kyoto University Mathematisches Institut University of Münster Article

First Online: 04 December 2013 Received: 02 October 2012 Accepted: 29 October 2013 DOI :
10.1007/s00209-013-1255-5

Cite this article as: Yokota, T. Math. Z. (2014) 277: 293. doi:10.1007/s00209-013-1255-5
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 Partly supported by JSPS Postdoctoral Fellowships for Research Abroad.

References 1.

Alexander, S., Kapovitch, V. Petrunin, A.: Alexandrov geometry. Forthcoming textbook

2.

Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence, RI (2001)

3.

Colding, T.: Large manifolds with positive Ricci curvature. Invent. Math.

124 (1–3), 193–214 (1996)

CrossRef MATH MathSciNet 4.

Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc.

1 , 443–474 (1979)

CrossRef MATH MathSciNet 5.

Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom.

18 (1), 1–147 (1983)

MATH MathSciNet 6.

Grove, K., Petersen, P.: A radius sphere theorem. Invent. Math.

112 (3), 577–583 (1993)

CrossRef MATH MathSciNet 7.

Grove, K., Wilhelm, F.: Hard and soft packing radius theorems. Ann. Math. (2)

142 (2), 213–237 (1995)

CrossRef MATH MathSciNet 8.

Kapovitch, V.: Perelman’s Stability Theorem. Surveys in Differential Geometry, vol. XI, pp. 103–136. International Press, Somerville, MA (2007)

9.

Katz, M.: The filling radius of two-point homogeneous spaces. J. Differ. Geom.

18 (3), 505–511 (1983)

MATH 10.

Mitsuishi, A.: A splitting theorem for infinite dimensional Alexandrov spaces with nonnegative curvature and its applications. Geom. Dedicata

144 , 101–114 (2010)

CrossRef MATH MathSciNet 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)

12.

Plaut, C.: Metric Spaces of Curvature \(\ge k\) . Handbook of Geometric Topology, pp. 818–898. North-Holland, Amsterdam (2002)

13.

Takatsu, A., Yokota, T.: Cone structure of

\(L^2\) -Wasserstein spaces. J. Topol. Anal.

4 (2), 237–253 (2012)

CrossRef MATH MathSciNet 14.

Wilhelm, F.: On the filling radius of positively curved manifolds. Invent. Math.

107 (3), 653–668 (1992)

CrossRef MATH MathSciNet 15.

Yamaguchi, T.: Simplicial volumes of Alexandrov spaces. Kyushu J. Math.

51 (2), 273–296 (1997)

CrossRef MATH MathSciNet 16.

Yokota, T.: A rigidity theorem in Alexandrov spaces with lower curvature bound. Math. Ann.

353 (2), 305–331 (2012)

CrossRef MATH MathSciNet 17.

Yokota, T.: On the filling radius of positively curved Alexandrov spaces. Math. Z.

273 (1–2), 161–171 (2013)

CrossRef MATH MathSciNet © Springer-Verlag Berlin Heidelberg 2013