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 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.