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

Abstract

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 $p,q\in M^n$ of total length $\le md$, where $m\in \{2,\dots ,(n+1)\}$ 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 $\frac{(n+1)(n+2)}{2}$ geodesic segments of total length $\le (n+1)(n+2)\mathrm{FillRad}{M}^n\le {(n+1)}^2{n}^n(n+2)\sqrt{(n+1)!}\mathrm{vol}{\left(M^n\right)}^{\frac{1}{n}}$.

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].