, Volume 14, Issue 4, pp 748-790

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

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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/4, 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 \( (n+2)! \textrm{FillRad}(M^n) \leq (n+2)!(n+1) n^n \sqrt{(n+1)!}(\textrm{vol}(M^n))^{1/n} \) where \( \textrm{FillRad}(M^n) \) where is the filling radius of a manifold, and \( (\textrm{vol}(M^n)) \) where is its volume.