{
  "abstract": "We study the relation between the saddle property and local metric minimality for smooth surfaces in Euclidean spaces. Anton Petrunin and Stephan Stadler proved that every smooth strictly saddle surface in $\\mathbb{R}^3$ is locally metric-minimising and expressed the expectation that this phenomenon fails in $\\mathbb{R}^4$. We confirm this expectation and prove a stronger density result: Every smooth strictly saddle embedding of a closed disc into $\\mathbb{R}^4$ can be approximated arbitrarily closely in the $C^\\infty$-topology by smooth strictly saddle embeddings that are nowhere locally metric-minimising. In particular, nowhere locally metric-minimising saddle embeddings exist in $\\mathbb{R}^d$ for every $d \\geq 4$. The main ingredient is the construction of non-trivial infinitesimal isometric deformations of saddle immersions in $\\mathbb{R}^4$ supported in arbitrarily small discs. We also provide an explicit twisted quadratic surface that is saddle in a neighbourhood of the origin but fails to be locally metric-minimising there, by deriving a necessary condition for metric minimality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10908",
  "author_total": 1,
  "authors": [
    {
      "name": "Stefan Christian Kohlmeier"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bdqsfdc0bvbvb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10908",
  "fulltext_url": null,
  "id": "bdqsfdc0bvbvb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10908",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7221164862",
  "pdf_url": "https://arxiv.org/pdf/2610.10908",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bdqsfdc0bvbvb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10908",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10908"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Nowhere Locally Metric-Minimising Saddle Surfaces",
  "title_text": "Nowhere Locally Metric-Minimising Saddle Surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/bdqsfdc0bvbvb",
  "venue": "arXiv",
  "versions": []
}