The Shortest Geodesic Flower on a Manifold with Locally Convex Ends and Finite Volume

TL;DR

This paper proves the existence of a shortest non-trivial geodesic net with specific bounds on edges and length on certain non-compact Riemannian manifolds with finite volume.

Contribution

It provides a quantitative version of a known result, establishing explicit bounds on geodesic nets on manifolds with convex ends and finite volume.

Findings

Existence of a non-trivial geodesic net with one vertex.

Bound on the number of edges: at most (n+2)(n+1)/2.

Total length bound: (n+2)(n+1)(n/2) * vol(M)^{1/n}.

Abstract

Suppose $M$ is a complete, non-compact $n$-dimensional Riemannian manifold with locally convex ends and finite volume. We prove that $M$ admits a non-trivial geodesic net with one vertex, at most $(n+2)(n+1)/2$ edges, and total length at most $$(n+2)(n+1)(n/2)\operatorname{vol}(M)^{1/n}.$$ This is a quantitative version of a result of G. R. Chambers, Y. Liokumovich, A. Nabutovsky and R. Rotman.