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 MnMn 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 mm geodesics connecting two points p,q∈Mnp,q∈Mn of total length ≤md≤md, where m∈{2,…,(n+1)}m∈{2,…,(n+1)} and dd is the diameter of MnMn. We also show that there exists a minimal geodesic net with at most n+1n+1 vertices and (n+1)(n+2)2 geodesic segments of total length ≤(n+1)(n+2)FillRadMn≤(n+1)2nn(n+2)(n+1)!vol(Mn)1n.These 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].