{"paper":{"title":"Strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A Lojasiewicz inequality for the pointed W-entropy establishes strong uniqueness of generalized cylindrical tangent flows in Ricci flow.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hanbing Fang, Yu Li","submitted_at":"2026-05-16T13:54:24Z","abstract_excerpt":"In this paper, we extend the results of \\cite{fang2025strong, fang2025singular} to generalized cylinders. More precisely, we establish a Lojasiewicz inequality for the pointed $\\mathcal{W}$-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder $\\mathbb{R}^k \\times N^{n-k}$, where $N$ is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset $\\mathcal{S}^k_{\\mathrm{qc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder R^k × N^{n-k}, where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset S^k_qc(N) is horizontally parabolic k-rectifiable.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The geometry near the base point is close to a generalized cylinder R^k × N^{n-k}, where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition (invoked in the statement of the Lojasiewicz inequality and its applications to uniqueness and rectifiability).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves Lojasiewicz inequality for W-entropy near generalized cylinders in Ricci flow, yielding strong uniqueness of tangent flows and horizontal parabolic k-rectifiability of the corresponding singularity set.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A Lojasiewicz inequality for the pointed W-entropy establishes strong uniqueness of generalized cylindrical tangent flows in Ricci flow.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"009a807ffe87f1d17b031f75f3b61d8804c0ea440dc4850762fa17b6563ccb56"},"source":{"id":"2605.17001","kind":"arxiv","version":1},"verdict":{"id":"876e60da-1760-4443-98f0-fc1e81ab8c37","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:42:34.010833Z","strongest_claim":"We establish a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder R^k × N^{n-k}, where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset S^k_qc(N) is horizontally parabolic k-rectifiable.","one_line_summary":"Proves Lojasiewicz inequality for W-entropy near generalized cylinders in Ricci flow, yielding strong uniqueness of tangent flows and horizontal parabolic k-rectifiability of the corresponding singularity set.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The geometry near the base point is close to a generalized cylinder R^k × N^{n-k}, where N is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition (invoked in the statement of the Lojasiewicz inequality and its applications to uniqueness and rectifiability).","pith_extraction_headline":"A Lojasiewicz inequality for the pointed W-entropy establishes strong uniqueness of generalized cylindrical tangent flows in Ricci flow."},"integrity":{"clean":false,"summary":{"advisory":0,"critical":1,"by_detector":{"doi_compliance":{"total":1,"advisory":0,"critical":1,"informational":0}},"informational":0},"endpoint":"/pith/2605.17001/integrity.json","findings":[{"note":"Identifier '10.1007/s10240-025-00145-3' is syntactically valid but the DOI registry (doi.org) returned 404, and Crossref / OpenAlex / internal corpus also have no record. 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