Flowers on Riemannian manifolds

Abstract

In this paper, we will present two upper bounds for the length of a smallest “flower-shaped” geodesic net in terms of the volume and the diameter of a manifold. Minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a Riemannian manifold. Let M ⁿ be a closed Riemannian manifold of dimension n. We prove that there exists a minimal geodesic net that consists of one vertex and at most 2n − 1 geodesic loops based at that vertex of total length ≤ 2n!d, where d is the diameter of M ⁿ. We also show that there exists a minimal geodesic net that consists of one vertex and at most ${3^{(n+1)^2}}$ loops of total length ${\leq2 (n+1)!^2 3^{(n+1)^3}\,Fill\,Rad\,M^n \leq2(n+1)!^{\frac{5}{2}}3^{(n+1)^3}(n+1)n^n vol(M^n)^{\frac{1}{n}}}$, where Fill Rad M ⁿ denotes the filling radius and vol(M ⁿ) denotes the volume of M ⁿ.