{
  "abstract": "We prove that Choe's higher-dimensional Enneper hypersurfaces in $\\mathbb R^{n+1}$, $3\\le n\\le6$, cannot be confined between two parallel hyperplanes, disproving his slab confinement conjecture. We further strengthen this conclusion by determining their precise height growth exponent. If $H(R)$ denotes the maximal absolute height above the asymptotic hyperplane within the horizontal ball of radius $R$, then $H(R)=R^{α_1+o(1)}$ as $R\\to\\infty$, where $α_1=\\frac{\\sqrt{(n-2)^2+4λ_1(Ω)}-(n-2)}{2}$, $Ω=S^{n-1}\\setminus[0,\\infty)^n$, and $λ_1(Ω)$ is the first Dirichlet eigenvalue of the spherical Laplacian. In particular, $0<α_1<1$, so the height is unbounded but grows sublinearly. The proof combines the blow-down behavior of a fundamental piece of Choe's hypersurface with the rigidity of positive harmonic functions on cones. In dimension two, the formula gives $2/3$, recovering the classical Enneper height-growth exponent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14343",
  "author_total": 1,
  "authors": [
    {
      "name": "Yike He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/75q4gemvarw1s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14343",
  "fulltext_url": null,
  "id": "75q4gemvarw1s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14343",
  "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": "W7221752061",
  "pdf_url": "https://arxiv.org/pdf/2610.14343",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/75q4gemvarw1s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14343",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14343"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Height of Choe's Higher-Dimensional Enneper Surface",
  "title_text": "The Height of Choe's Higher-Dimensional Enneper Surface",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/75q4gemvarw1s",
  "venue": "arXiv",
  "versions": []
}