{
  "abstract": "Alpöge recently announced smooth counterexamples to the Loewner and Carathéodory conjectures. We present a simpler construction based on a trigonometric function as a seed. Using this, for every positive integer $k$, we construct a smooth real-valued function on the plane, flat at the origin, whose Hessian eigenline fields have an isolated singularity of index $1+k/2$. A support-function parametrization realizes this index as an umbilic index of a non-real analytic $C^\\infty$-regular surface germ in Euclidean $3$-space. The same framework, applied to a rapidly decreasing function on the whole plane, gives a strictly convex closed smooth surface with exactly one umbilic, of index $2$. This paper is self-contained with regard to the existence of an umbilic of index $2$ and the construction of a counterexample to the Carathéodory conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34036",
  "author_total": 2,
  "authors": [
    {
      "name": "Naoya Ando"
    },
    {
      "name": "Masaaki Umehara"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/59cgrfgvppvcy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34036",
  "fulltext_url": null,
  "id": "59cgrfgvppvcy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34036",
  "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"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7223898855",
  "pdf_url": "https://arxiv.org/pdf/2610.34036",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/59cgrfgvppvcy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34036",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34036"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A simple construction of smooth counterexamples to the Loewner and Carathéodory conjectures",
  "title_text": "A simple construction of smooth counterexamples to the Loewner and Carathéodory conjectures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/59cgrfgvppvcy",
  "venue": "arXiv",
  "versions": []
}