Geodesic nets on non-compact Riemannian manifolds

Abstract

Abstract A geodesic flower is a finite collection of geodesic loops based at the same point 𝑝 that satisfy the following balancing condition: the sum of all unit tangent vectors to all geodesic arcs meeting at 𝑝 is equal to the zero vector. In particular, a geodesic flower is a stationary geodesic net. We prove that, in every complete non-compact manifold with locally convex ends, there exists a non-trivial geodesic flower.