Geodesic nets on non-compact Riemannian manifolds

TL;DR

This paper proves the existence of non-trivial geodesic flowers, a special type of stationary geodesic net, on complete non-compact Riemannian manifolds with locally convex ends.

Contribution

It establishes the existence of geodesic flowers on a broad class of non-compact manifolds, extending understanding of geodesic nets in geometric analysis.

Findings

Existence of non-trivial geodesic flowers on manifolds with convex ends

Geodesic flowers are stationary geodesic nets satisfying a balancing condition

Applicable to all complete non-compact manifolds with locally convex ends

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.