{
  "abstract": "We study large-data continuation below the critical energy for the two-dimensional sphere-valued Schrödinger map \\[ \\partial_t u=u\\timesΔu, \\qquad u:[0,T_+)\\times\\mathbb{R}^2\\to\\mathbb{S}^2, \\qquad E(u_0)<4π. \\] Working directly with the original equation through the complex tangential fields $W_m=\\partial_m u+\\mathrm{i} u\\times\\partial_m u$ and the ambient connections $Γ_α$, we prove frequency-envelope bounds governed by the critical norm $\\|\\nabla u\\|_{\\ell^2L^4_{t,x}}$. As consequences, $T_+=\\infty$ if $\\|\\nabla u\\|_{\\ell^2L^4([0,T_+)\\times\\mathbb{R}^2)}<\\infty$, and also if a terminal one-window high-frequency tail $\\mathcal{A}_{\\mathrm{win}}(t)$ satisfies $\\liminf_{t\\uparrow T_+}\\mathcal{A}_{\\mathrm{win}}(t)<\\varepsilon_*$. Thus any finite-time failure must force persistent high-frequency concentration on all sufficiently late critical parabolic windows.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02721",
  "author_total": 2,
  "authors": [
    {
      "name": "Pang-Hung Chung"
    },
    {
      "name": "Dan Han"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/38knnprf8j1wf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02721",
  "fulltext_url": null,
  "id": "38knnprf8j1wf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02721",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7220427244",
  "pdf_url": "https://arxiv.org/pdf/2610.02721",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/38knnprf8j1wf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02721",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02721"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Large-Data Global Criteria for Two-Dimensional Critical-Energy Schrödinger Maps",
  "title_text": "Large-Data Global Criteria for Two-Dimensional Critical-Energy Schrödinger Maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/38knnprf8j1wf",
  "venue": "arXiv",
  "versions": []
}