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Second, we prove that if the conformally flat hypersurface with constant M\\\"obius scalar curvature $R$ is compact, then $$R=(n-1)(n-2)r^2, ~~0<r<1,$$ and the compact conformally flat hypersurface is M\\\"obius equivalent to the torus $$\\mathbb{ S}^1(\\sq"},"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":"1709.01658","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2017-09-06T03:01:17Z","cross_cats_sorted":[],"title_canon_sha256":"ae7be3ae0daed0fffbca7cc2afdfca39e4f8e66b60dc462d01b1887d440553ac","abstract_canon_sha256":"66eb35c76f23ed35677b55c2c4f2cda440e3c3f1aac56fb86aef24a4b1b3337d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:54.669720Z","signature_b64":"Twc//SgEo+xIvIjy/5aZLib7lzPv5ptDJ9XONov7V2sk8KpA5DraJtjhIytgzO2zVBm70rwnlJ7uuMWqMooCDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93c769e7c4c3760cbd1ad4dc200933d9962dae6287d677717ad030ef5d111692","last_reissued_at":"2026-05-18T00:35:54.669085Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:54.669085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A M\\\"obius scalar curvature rigidity on compact conformally flat hypersurfaces in $\\mathbb{S}^{n+1}$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Changping Wang, Limiao Lin, Tongzhu Li","submitted_at":"2017-09-06T03:01:17Z","abstract_excerpt":"In this paper, we study conformally flat hypersurfaces of dimension $n(\\geq 4)$ in $\\mathbb{S}^{n+1}$ using the framework of M\\\"obius geometry. 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