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In particular, we show that $${\\rm Vol}_{g(t)} \\geq e^{ T\\lambda-\\frac{n}{2}} \\left(\\frac{4}{(A(\\lambda-r)+4B)T}\\right)^{\\frac{n}{2}}\\left(T-t\\right)^{\\frac{n}{2}},$$ where $r:=\\inf_{\\|\\phi\\|_2^2=1} \\int_M R\\phi^2 \\ d{\\rm vol}_{g(0)}$, $\\lambda:=\\inf_{\\|\\phi\\|_2^2=1} \\int_M 4|\\nabla\\phi|^2+R\\phi^2\\ d{\\rm vol}_{g(0)}$ and $A"},"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":"1803.09591","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-03-26T13:50:38Z","cross_cats_sorted":[],"title_canon_sha256":"6b394740db83bc08d8612fde7c3323abc04c0febacc415a460be3b86e2dd6ea8","abstract_canon_sha256":"e1a28ac28f20ddc3e373bb9579256041b0d8b74f323f60221a211800ef32144e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:10.982848Z","signature_b64":"GrjxLxLyih89drC8ncFNODCVtQCPU4RtOW/ADq5R6kVLrOJB6wXYWqxEAPAOddlS8Ldgb58Ek3UzKznEH5SKAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d47388ab7ca20f9caf38593bcc85cf38428395e6ce624a18337ddf9a9a85969","last_reissued_at":"2026-05-18T00:20:10.982094Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:10.982094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Volume bounds of the Ricci flow on closed manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Chih-Wei Chen, Zhenlei Zhang","submitted_at":"2018-03-26T13:50:38Z","abstract_excerpt":"Let $\\{g(t)\\}_{t\\in [0,T)}$ be the solution of the Ricci flow on a closed Riemannian manifold $M^n$ with $n\\geq 3$. Without any assumption, we derive lower volume bounds of the form ${\\rm Vol}_{g(t)}\\geq C (T-t)^{\\frac{n}{2}}$, where $C$ depends only on $n$, $T$ and $g(0)$. In particular, we show that $${\\rm Vol}_{g(t)} \\geq e^{ T\\lambda-\\frac{n}{2}} \\left(\\frac{4}{(A(\\lambda-r)+4B)T}\\right)^{\\frac{n}{2}}\\left(T-t\\right)^{\\frac{n}{2}},$$ where $r:=\\inf_{\\|\\phi\\|_2^2=1} \\int_M R\\phi^2 \\ d{\\rm vol}_{g(0)}$, $\\lambda:=\\inf_{\\|\\phi\\|_2^2=1} \\int_M 4|\\nabla\\phi|^2+R\\phi^2\\ d{\\rm vol}_{g(0)}$ and $A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09591","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1803.09591","created_at":"2026-05-18T00:20:10.982221+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.09591v1","created_at":"2026-05-18T00:20:10.982221+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.09591","created_at":"2026-05-18T00:20:10.982221+00:00"},{"alias_kind":"pith_short_12","alias_value":"RVDTRCVXZIQP","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_16","alias_value":"RVDTRCVXZIQPTSXT","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_8","alias_value":"RVDTRCVX","created_at":"2026-05-18T12:32:50.500415+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RVDTRCVXZIQPTSXTQWJ3ZSC46O","json":"https://pith.science/pith/RVDTRCVXZIQPTSXTQWJ3ZSC46O.json","graph_json":"https://pith.science/api/pith-number/RVDTRCVXZIQPTSXTQWJ3ZSC46O/graph.json","events_json":"https://pith.science/api/pith-number/RVDTRCVXZIQPTSXTQWJ3ZSC46O/events.json","paper":"https://pith.science/paper/RVDTRCVX"},"agent_actions":{"view_html":"https://pith.science/pith/RVDTRCVXZIQPTSXTQWJ3ZSC46O","download_json":"https://pith.science/pith/RVDTRCVXZIQPTSXTQWJ3ZSC46O.json","view_paper":"https://pith.science/paper/RVDTRCVX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.09591&json=true","fetch_graph":"https://pith.science/api/pith-number/RVDTRCVXZIQPTSXTQWJ3ZSC46O/graph.json","fetch_events":"https://pith.science/api/pith-number/RVDTRCVXZIQPTSXTQWJ3ZSC46O/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RVDTRCVXZIQPTSXTQWJ3ZSC46O/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RVDTRCVXZIQPTSXTQWJ3ZSC46O/action/storage_attestation","attest_author":"https://pith.science/pith/RVDTRCVXZIQPTSXTQWJ3ZSC46O/action/author_attestation","sign_citation":"https://pith.science/pith/RVDTRCVXZIQPTSXTQWJ3ZSC46O/action/citation_signature","submit_replication":"https://pith.science/pith/RVDTRCVXZIQPTSXTQWJ3ZSC46O/action/replication_record"}},"created_at":"2026-05-18T00:20:10.982221+00:00","updated_at":"2026-05-18T00:20:10.982221+00:00"}