{
  "abstract": "For each integer $k\\geq 1$, we study the $(k+1)$-dimensional helicoid $H_k\\subset\\mathbb{R}^{2k+1}$ parametrized by \\[ (u_1,\\ldots,u_k,s) \\longmapsto \\bigl(u_1e^{is},\\ldots,u_ke^{is},s\\bigr) \\in \\mathbb{C}^k\\times\\mathbb{R}\\cong \\mathbb{R}^{2k+1}. \\] These helicoids form a basic and distinguished family of complete, properly embedded minimal submanifolds diffeomorphic to $\\mathbb{R}^{k+1}$, and provide natural higher-dimensional analogues of the classical helicoid in $\\mathbb{R}^3$. We completely determine their stability: $H_k$ is stable for $k\\geq 3$ and unstable for $k\\leq 2$. The sharp transition at $k=3$ is particularly striking: while the classical helicoid $(k=1)$ and its first higher-dimensional analogue $(k=2)$ are unstable, the four-dimensional helicoid $H_3\\subset\\mathbb{R}^7$ is already stable. For $k\\geq 3$, we also determine their area-minimizing property: $H_k$ is area-minimizing when $k$ is even and not area-minimizing when $k$ is odd. The area-minimizing result is proved by constructing an explicit calibration, while the non-area-minimizing result follows from an explicit competitor. In particular, for every even $k\\geq 4$, the $(k+1)$-dimensional helicoid $H_k$ is an entire minimal graph in $\\mathbb{R}^{2k+1}$ that is area-minimizing.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11481",
  "author_total": 4,
  "authors": [
    {
      "name": "Chung-Jun Tsai"
    },
    {
      "name": "Mao-Pei Tsui"
    },
    {
      "name": "Jingbo Wan"
    },
    {
      "name": "Mu-Tao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4zze41btn89vn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11481",
  "fulltext_url": null,
  "id": "4zze41btn89vn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11481",
  "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",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212226387",
  "pdf_url": "https://arxiv.org/pdf/2609.11481",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4zze41btn89vn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11481",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11481"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Stability and Area-Minimizing Property of Higher-Dimensional Helicoids",
  "title_text": "Stability and Area-Minimizing Property of Higher-Dimensional Helicoids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/4zze41btn89vn",
  "venue": "arXiv",
  "versions": []
}