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We show that the obstruction to extend the Ricci flow is the boundedness $L^{3/2}$-norm of the scalar curvature $R(t)$, i.e, the Ricci flow can be extended over time $T$ if and only if the $||R(t)||_{L^{3/2}}$ is uniformly bounded for $0 \\leq t < T$ . On the other hand, if the fundamental group of $M$ is finite and the $||R(t)||_{L^{\\3/2}}$ is bounded for all time under the Ricci flow, then $M$ is diffeomorphic to a 3-dimensional spherical space-form."},"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":"1101.0637","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-01-04T02:02:11Z","cross_cats_sorted":[],"title_canon_sha256":"5359ca6c17bb300a813029cc28ae496afb9dea1f0d23319b790b6b01bd051374","abstract_canon_sha256":"5e54176bc877ba23bde06259da2b7e66b1a933c4aae2c3f1fad636bbe6f6d036"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:47.303897Z","signature_b64":"21TaDlonF5YTdjScz0mWG17UAcigpXgUrRq9jybynydLcS9AHOX2PC8ZYDjiqbYdXQ3fsMtZjkE4Ady8DL9VDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e48c04063073da5d6325b3ca19f894493960d10eda85310ec53e0076bc9d2bb","last_reissued_at":"2026-05-18T04:27:47.303288Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:47.303288Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $L^{3/2}$-norm of the scalar curvature under the Ricci flow on a 3-manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongnian Huang","submitted_at":"2011-01-04T02:02:11Z","abstract_excerpt":"Assume $M$ is a closed 3-manifold whose universal covering is not $S^3$. 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