The length of a shortest geodesic net on a closed Riemannian manifold

In this paper we will estimate the smallest length of a minimal geodesic net on an arbitrary closed Riemannian manifold M n in terms of the diameter of this manifold and its dimension. Minimal geodesic nets are critical points of the length functional on the space of immersed graphs into a Riemannian manifold. We prove that there exists a minimal geodesic net that consists of m geodesics connecting two points p , q ∈ M n of total length ≤ m d , where m ∈ { 2 , … , ( n + 1 ) } and d is the diameter of M n . We also show that there exists a minimal geodesic net with at most n + 1 vertices and ( n + 1 ) ( n + 2 ) 2 geodesic segments of total length ≤ ( n + 1 ) ( n + 2 ) FillRad M n ≤ ( n + 1 ) 2 n n ( n + 2 ) ( n + 1 ) ! vol ( M n ) 1 n . results significantly improve one of the results of [A. Nabutovsky, R. Rotman, The minimal length of a closed geodesic net on a Riemannian manifold with a nontrivial second homology group, Geom. Dedicata 113 (2005) 234–254] as well as most of the results of [A. Nabutovsky, R. Rotman, Volume, diameter and the minimal mass of a stationary 1-cycle, Geom. Funct. Anal. 14 (4) (2004) 748–790].