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Based on this key estimate, we further obtain compactness of the base warping functions $u_i"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.25116","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2026-05-24T15:02:38Z","cross_cats_sorted":[],"title_canon_sha256":"c17c8c101455ccf71182ef0a923f1be4b7e84890653011e8b3cad3bc82183e44","abstract_canon_sha256":"17e4b4e46c6f1305ac4f31482741a963dd74a836313f840b568f9846569ad442"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:04:18.780244Z","signature_b64":"ZtvWGOoziiqJldLVQHmrvpx5Nto2fF0Mw+qn/QU0Hi3821iSTn3TOXDmmprBPFgPm2GHvk44mXbX1ylB42TuBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e8a6d19e58ec1a5a5e3d711eb22c14655c3649825e7d943fabddeb6bc774b4a","last_reissued_at":"2026-05-26T02:04:18.779442Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:04:18.779442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scalar Curvature Compactness for Warped Products on $\\mathbb{S}^2\\times\\mathbb{S}^1$ with Varying Base Metrics","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Changliang Wang, Zhixin Wang","submitted_at":"2026-05-24T15:02:38Z","abstract_excerpt":"We study the Gromov--Sormani MinA scalar curvature compactness conjecture for warped product metrics on $\\mathbb{S}^2\\times\\mathbb{S}^1$ of the form introduced by Kazaras-Xu in \\cite{KazarasXu2023} as follows: \\[ g_i=\\varphi_i^{-2}h_i+\\varphi_i^2d\\xi^2, \\qquad h_i=dr^2+u_i^2(r)d\\theta^2. \\] Assuming nonnegative scalar curvature, a uniform volume upper bound, and a positive lower bound for the areas of closed minimal surfaces, we prove a uniform diameter bound for the base surfaces $(\\mathbb{S}^2,h_i)$. 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