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

Fuente: arXiv

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Autore principale:	Nguyen, Duc Toan
Natura:	Preprint
Pubblicazione:	2024
Soggetti:	Differential Geometry
Accesso online:	Acceder al recurso
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_version_	1866914072805507072
author	Nguyen, Duc Toan
author_facet	Nguyen, Duc Toan
contents	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.
format	Preprint
id	arxiv_https___arxiv_org_abs_2412_02872
institution	arXiv
publishDate	2024
record_format	arxiv
spellingShingle	On the existence of a balanced vertex in geodesic nets with three boundary vertices Nguyen, Duc Toan Differential Geometry 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.
title	On the existence of a balanced vertex in geodesic nets with three boundary vertices
topic	Differential Geometry
url	https://arxiv.org/abs/2412.02872

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