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PreprintCC BY-NC-ND 4.0arXiv

Geodesic nets on the Euclidean plane and closed geodesic nets on Euclidean surfaces

  • Ivan Frolov
  • Denis Gorodkov
  • Dmitrii Korshunov
  • Alexander Nabutovsky
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Abstract

We prove that if M is a closed Riemannian surface of diameter d and area v with sectional curvature in the [−1,1] interval, then a closed geodesic net of length l has at most f(l,d,v) branch points, where f(l,d,v)=(400l̅)(180l̅)4 for l̅= max{l, exp(d)/(min{1, v/4})} This answers a question posed by S. Becker-Kahn. We also prove that for each geodesic net in the Euclidean plane with at most n unbalanced (boundary) vertices such that all its unbalanced vertices have degree 1, the number of balanced vertices of degree ≥ 3 (=branch points) does not exceed (25n)2n2. This answers a question posed in [GM] and [NP].

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Cite

BibTeXRIS
BibTeX
@misc{frolov2026geodesic,
  title = {{Geodesic nets on the Euclidean plane and closed geodesic nets on Euclidean surfaces}},
  author = {Ivan Frolov and Denis Gorodkov and Dmitrii Korshunov and Alexander Nabutovsky},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.14080},
  eprint = {2606.14080},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.14080},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Geodesic nets on the Euclidean plane and closed geodesic nets on Euclidean surfaces
AU  - Ivan Frolov
AU  - Denis Gorodkov
AU  - Dmitrii Korshunov
AU  - Alexander Nabutovsky
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2606.14080
UR  - https://arxiv.org/abs/2606.14080
AB  - We prove that if M is a closed Riemannian surface of diameter d and area v with sectional curvature in the [−1,1] interval, then a closed geodesic net of length l has at most f(l,d,v) branch points, where f(l,d,v)=(400l̅)^(180l̅)^4 for l̅= max{l, exp(d)/(min{1, v/4})} This answers a question posed by S. Becker-Kahn. We also prove that for each geodesic net in the Euclidean plane with at most n unbalanced (boundary) vertices such that all its unbalanced vertices have degree 1, the number of balanced vertices of degree ≥ 3 (=branch points) does not exceed (25n)^2n^2. This answers a question posed in [GM] and [NP].
ER  -