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

Duc Toan Nguyen

TL;DR

The paper investigates the existence of a balanced vertex in geodesic nets with three boundary vertices on 2D Riemannian surfaces, generalizing the Fermat point from Euclidean geometry. It develops a continuity-based construction, leveraging a Y_X path along geodesic edges and a 2π/3 angle condition, supplemented by Jacobi-field and comparison-geometry tools to propagate angle increases. The main result shows that on any surface with Gaussian curvature bounded above by $1/R^2$, a triangle with all angles $< 2\pi/3$ possesses a balanced point provided the triangle's diameter is under $R\pi/2$; a corresponding sphere case is analyzed, and counterexamples illustrate the role of curvature and diameter bounds. These findings connect classical Fermat-point theory with geodesic nets on curved surfaces, offering explicit sufficient conditions for balanced vertices and enriching the theory of stationary geodesic nets. The work has implications for understanding minimal-net structures and their stationary properties on general Riemannian manifolds.

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 $2π/3$, 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.

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

TL;DR

The paper investigates the existence of a balanced vertex in geodesic nets with three boundary vertices on 2D Riemannian surfaces, generalizing the Fermat point from Euclidean geometry. It develops a continuity-based construction, leveraging a Y_X path along geodesic edges and a 2π/3 angle condition, supplemented by Jacobi-field and comparison-geometry tools to propagate angle increases. The main result shows that on any surface with Gaussian curvature bounded above by , a triangle with all angles possesses a balanced point provided the triangle's diameter is under ; a corresponding sphere case is analyzed, and counterexamples illustrate the role of curvature and diameter bounds. These findings connect classical Fermat-point theory with geodesic nets on curved surfaces, offering explicit sufficient conditions for balanced vertices and enriching the theory of stationary geodesic nets. The work has implications for understanding minimal-net structures and their stationary properties on general Riemannian manifolds.

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 , 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.

Paper Structure

This paper contains 6 sections, 20 theorems, 56 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminary Background
  3. Existence of a balanced vertex in geodesic nets with three boundary vertices
  4. General conditions for the existence of a balanced vertex
  5. Existence of a balanced point on a surface with non-positive curvature
  6. Existence of a balanced point on a general surface

Key Result

Theorem 1.1

Let $M$ be a Riemannian surface such that its Gaussian curvature is bounded above by $1/R^2$, for $R>0$. Let triangle $ABC$ on $M$ be given such that its three angles measure less than $2\pi/3$. If the maximum geodesic distance of two points in the domain of the triangle $ABC$ is less than $R\pi/2$,

Figures (11)

  • Figure 1: Angles
  • Figure 2: Existence of $Y$ with $m(\angle BYC)=\dfrac{2\pi}{3}$ for every $X$
  • Figure 3: Existence of balanced point
  • Figure 5: Condition for continuity of $Y_X$
  • Figure 6: Existence of a balanced point in non-positive-curvature case
  • ...and 6 more figures

Theorems & Definitions (41)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Gauss-Bonnet Kobayashi2019
  • Corollary 2.4
  • Definition 2.5
  • Theorem 2.6: The Lebesgue number lemma MR464128
  • Definition 2.7
  • Lemma 2.8
  • ...and 31 more