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

الأصل · EN

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/3, 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 2(n+2)!Fill Rad(Mⁿ) and, as a corollary, by 2(n+2)!(n+1)nⁿ(n!)¹/²(vol(Mⁿ))¹/ⁿ, where Fill Rad(Mⁿ) is the filling radius of the manifold, and vol(Mⁿ) is its volume.