The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group

Abstract

Let M n be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on M n of length at most 3 diameter(M n ). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(M n ), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of M n . If n=2, then this critical θ-graph is not only immersed but embedded.