Singular sets in noncollapsed Ricci flow limit spaces

[2510.26317] Singular sets in noncollapsed Ricci flow limit spaces Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2510.26317 (math) [Submitted on 30 Oct 2025 ( v1 ), last revised 15 Aug 2026 (this version, v3)] Title: Singular sets in noncollapsed Ricci flow limit spaces Authors: Hanbing Fang , Yu Li View a PDF of the paper titled Singular sets in noncollapsed Ricci flow limit spaces, by Hanbing Fang and 1 other authors View PDF HTML (experimental) Abstract: In this paper, we study the singular set $\mathcal{S}$ of a noncollapsed Ricci flow limit space, arising as the pointed Gromov--Hausdorff limit of a sequence of closed Ricci flows with uniformly bounded entropy. The singular set $\mathcal{S}$ admits a natural stratification: \begin{equation*} \mathcal S^0 \subset \mathcal S^1 \subset \cdots \subset \mathcal S^{n-2}=\mathcal S, \end{equation*} where a point $z \in \mathcal S^k$ if and only if no tangent flow at $z$ is $(k+1)$-symmetric. In general, the Hausdorff dimension of $\mathcal S^k$ with respect to the spacetime distance is at most $k$. We show that the subset $\mathcal{S}^k_{\mathrm{qc}} \subset \mathcal{S}^k$, consisting of points where some tangent flow is given by a standard cylinder or its quotient, is parabolic $k$-rectifiable. In dimension four, we prove the stronger statement that each stratum $\mathcal{S}^k$ is parabolic $k$-rectifiable for $k \in \{0, 1, 2\}$. Furthermore, we establish a sharp uniform $\mathscr{H}^2$-volume bound for $\mathcal{S}$ and show that, up to a set of $\mathscr{H}^2$-measure zero, the tangent flow at any point in $\mathcal{S}$ is backward unique. In addition, we derive $L^1$-curvature bounds for four-dimensional closed Ricci flows. As an application, we resolve Perelman's bounded diameter conjecture for three-dimensional closed Ricci flows. Comments: Revised version. The spacetime distance has been updated in accordance with arXiv:2510.12398v3 , and several typos have been corrected. All main results remain unchanged Subjects: Differential Geometry (math.DG) Cite as: arXiv:2510.26317 [math.DG] (or arXiv:2510.26317v3 [math.DG] for this version) https://doi.org/10.48550/arXiv.2510.26317 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Yu Li [ view email ] [v1] Thu, 30 Oct 2025 10:11:49 UTC (99 KB) [v2] Thu, 9 Apr 2026 09:46:35 UTC (103 KB) [v3] Sat, 15 Aug 2026 02:29:16 UTC (104 KB) Full-text links: Access Paper: View a PDF of the paper titled Singular sets in noncollapsed Ricci flow limit spaces, by Hanbing Fang and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2025-10 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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