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

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 M^n is bounded from above by (n+2)!d/3, where d is the diameter of a manifold M^n. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold M^n is bounded from above by 2(n+2)!Fill Rad(M^n) and, as a corollary, by 2(n+2)!(n+1)n^n(n!)^{1/2}(vol(M^n))^{1/n}, where Fill Rad(M^n) is the filling radius of the manifold, and vol(M^n) is its volume.