{
  "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 $f(l,d,v)=(400\\bar{l})^{(180\\bar{l})^4}$ for $\\bar{l}= \\max\\{l, \\frac{\\exp(d)}{\\min\\{1, \\frac{v}{4}\\}}\\}$ 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 $\\ge 3$ (=branch points) does not exceed $(25n)^{2n^2}$. This answers a question posed in [GM] and [NP].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.14080",
  "author_total": 4,
  "authors": [
    {
      "name": "Ivan Frolov"
    },
    {
      "name": "Denis Gorodkov"
    },
    {
      "name": "Dmitrii Korshunov"
    },
    {
      "name": "Alexander Nabutovsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8j5v2ask6h9p7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.14080",
  "fulltext_url": null,
  "id": "8j5v2ask6h9p7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.14080",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7164856048",
  "pdf_url": "https://arxiv.org/pdf/2606.14080",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8j5v2ask6h9p7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2606.14080",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.14080"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Geodesic nets on the Euclidean plane and closed geodesic nets on Euclidean surfaces",
  "title_text": "Geodesic nets on the Euclidean plane and closed geodesic nets on Euclidean surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/8j5v2ask6h9p7",
  "venue": "arXiv",
  "versions": []
}