{
  "abstract": "For a map $F:Ω\\subset\\mathbb{R}^n\\to\\mathbb{R}^m$, let $Θ(F)$ be the $n$-form on $Ω\\times\\mathbb{R}^m$ introduced by Tsai and Wang [arXiv:2604.04336]. If $λ_1,\\ldots,λ_{r}$ are the nonzero singular values of $d F$ at any fixed point, we prove that $$ Θ(F) \\text{ has comass one if and only if }~ \\mathcal{S}(λ):=\\sum_{i=1}^{r}\\frac{λ_i^2}{1+λ_i^2}\\leq1. $$ Note that if $\\text{rank} d F\\le1$, $\\mathcal{S}$ is always less than $1$. We also prove a dichotomy that if $\\mathcal{S}\\leq 1$, then either $\\mathcal{S} < 1$ or $\\mathcal{S}\\equiv1$. When $\\mathcal{S}<1$, the graph tangent plane is the only calibrated plane, generalizing the corresponding result for hypersurfaces. When $\\mathcal{S}\\equiv1$, we prove that $F$ is affine or of rank $2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39159",
  "author_total": 2,
  "authors": [
    {
      "name": "Wei-Chun Chen"
    },
    {
      "name": "Chun-Kai Lien"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/08dtjcgzv5v8e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39159",
  "fulltext_url": null,
  "id": "08dtjcgzv5v8e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39159",
  "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": "W7215124805",
  "pdf_url": "https://arxiv.org/pdf/2609.39159",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/08dtjcgzv5v8e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39159",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39159"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Exact Comass Criterion for the Tsai--Wang Form",
  "title_text": "The Exact Comass Criterion for the Tsai--Wang Form",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/08dtjcgzv5v8e",
  "venue": "arXiv",
  "versions": []
}