{
  "abstract": "In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed $G_2$-structures. Under a lower scalar-curvature bound and a distance-dependent bound on the gradient of the soliton potential, we prove pointed measured Gromov-Hausdorff compactness. With a uniform lower bound for the localised Perelman entropy, the Gromov-Hausdorff convergence improves to pointed $C^{1,α}$ convergence. Our principal result shows that this \\(C^{1,α}\\) convergence upgrades to smooth convergence on the regular set. More precisely, after passing to a subsequence, the metrics, the defining positive \\(3\\)-forms, and the soliton potentials converge smoothly on every compact subset of the regular set, and the limiting data define a gradient Laplacian soliton. The proof develops a local entropy method adapted to \\(G_2\\)-solitons. Since the available \\(C^{1,α}\\) control does not directly close an elliptic bootstrap for the \\(G_2\\)-soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy \\(\\varepsilon\\)-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed \\(C^{1,α}\\) convergence gives an almost-Euclidean local isoperimetric inequality, which in turn verifies the required small-entropy condition automatically. The resulting curvature bounds are then combined with \\(G_2\\)-specific differential identities and quantitative interior estimates to control the soliton data. Finally, at the critical exponent in dimension seven, we show that a uniform weighted \\(L^{\\frac{7}{2}}\\) -curvature bound then yields pointed $C^\\infty$ compactness.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.06619",
  "author_total": 3,
  "authors": [
    {
      "name": "Haozhao Li"
    },
    {
      "name": "Yuanqing Ma"
    },
    {
      "name": "Kai Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cm3d2vz28r6k9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.06619",
  "fulltext_url": null,
  "id": "cm3d2vz28r6k9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.06619",
  "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.DG"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7163888205",
  "pdf_url": "https://arxiv.org/pdf/2606.06619",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cm3d2vz28r6k9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2606.06619",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.06619"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the structure of complete $G_2$-solitons",
  "title_text": "On the structure of complete G_2-solitons",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/cm3d2vz28r6k9",
  "venue": "arXiv",
  "versions": []
}