Flowers on Riemannian manifolds

In this paper, we will present two upper bounds for the length of a smallest "flower-shaped" geodesic net in terms of the volume and the diameter of a manifold. Minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a Riemannian manifold. Let M-n be a closed Riemannian manifold of dimension n. We prove that there exists a minimal geodesic net that consists of one vertex and at most 2n-1 geodesic loops based at that vertex of total length <= 2n!d, where d is the diameter of M-n. We also show that there exists a minimal geodesic net that consists of one vertex and at most 3((n+1)2) loops of total length <= 2(n + 1)!(2)3((n+ 1)3) FillRadM(n) <= 2(n + 1)!(5/2)3((n+1)3) (n + 1)n(n) vol(M-n)(1/n), where FillRadMn denotes the filling radius and vol(M-n) denotes the volume of Mn.