Geometric & Functional Analysis GAFA

, Volume 14, Issue 4, pp 748–790

Volume, diameter and the minimal mass of a stationary 1-cycle

Original Article

DOI: 10.1007/s00039-004-0474-7

Cite this article as:
Nabutovsky, A. & Rotman, R. Geom. funct. anal. (2004) 14: 748. doi:10.1007/s00039-004-0474-7

Abstract

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold Mn is bounded from above by (n + 2)!d/4, where d is the diameter of a manifold Mn. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold Mn is bounded from above by \( (n+2)! \textrm{FillRad}(M^n) \leq (n+2)!(n+1) n^n \sqrt{(n+1)!}(\textrm{vol}(M^n))^{1/n} \) where \( \textrm{FillRad}(M^n) \) where is the filling radius of a manifold, and \( (\textrm{vol}(M^n)) \) where is its volume.

Keywords.

((no keywords given))

Mathematics Subject Classification (2000).

((no classification given))

Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada