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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2\pi /3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>π</mi> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年10月3日

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  • Duc Toan Nguyen

摘要

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 \(2\pi /3\) 2 π / 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.