On the existence of a balanced vertex in geodesic nets with three boundary vertices

TL;DR

This paper proves the existence of a balanced vertex in geodesic nets with three boundary vertices on certain Riemannian surfaces, generalizing the Fermat point concept to curved spaces.

Contribution

It establishes conditions under which a balanced vertex exists in geodesic nets on Riemannian surfaces, extending classical Euclidean results.

Findings

Existence of a balanced vertex when all angles are less than 2π/3.

Conditions on side lengths for the existence of the balanced vertex.

Generalization of the Fermat point to curved surfaces.

Abstract

Geodesic nets are types of graphs in Riemannian manifolds where each edge is a geodesic segment. One important object used in the construction of geodesic nets is a balanced vertex, where the sum of unit tangent vectors along adjacent edges is zero. We prove the existence of a balanced vertex of a triangle (with three unbalanced vertices) on a general two-dimensional Riemannian surface when all angles measure less than $2\pi/3$, if the length of the sides of the triangle is not too large. This property is a generalization for the existence of the Fermat point of a planar triangle.