{
  "abstract": "Let $(X,d,p)$ be a nontrivial pointed Gromov--Hausdorff limit of complete connected Riemannian $n$-manifolds satisfying $\\operatorname{Ric}\\geq-(n-1)g$. For $1\\leq j\\leq n$, set \\[ \\mathcal{R}_j(X) = \\bigl\\{x\\in X:\\text{ every tangent cone at } x \\text{ is isometric to } \\mathbb{R}^j\\bigr\\}, \\qquad \\mathcal{R}(X)=\\bigsqcup_{j=1}^n\\mathcal{R}_j(X). \\] We prove that there is a unique $k$ such that $\\mathcal{R}(X)=\\mathcal{R}_k(X)$, answering the pointwise regular-dimension question originating in Cheeger and Colding's work~\\cite[p408]{CC97}. We construct a volume-corrected heat entropy on RCD space and identify the distributional logarithmic-density defect $\\operatorname{tr}(\\operatorname{Hess} f)-Δf$ by Green-volume regularization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41449",
  "author_total": 1,
  "authors": [
    {
      "name": "Tianyin Ren"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e50e94rhh44p8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41449",
  "fulltext_url": null,
  "id": "e50e94rhh44p8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41449",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41449",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e50e94rhh44p8.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41449",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41449"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The regular set of Ricci limit spaces",
  "title_text": "The regular set of Ricci limit spaces",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/e50e94rhh44p8",
  "venue": "arXiv",
  "versions": []
}