{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:RVDTRCVXZIQPTSXTQWJ3ZSC46O","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e1a28ac28f20ddc3e373bb9579256041b0d8b74f323f60221a211800ef32144e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-03-26T13:50:38Z","title_canon_sha256":"6b394740db83bc08d8612fde7c3323abc04c0febacc415a460be3b86e2dd6ea8"},"schema_version":"1.0","source":{"id":"1803.09591","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.09591","created_at":"2026-05-18T00:20:10Z"},{"alias_kind":"arxiv_version","alias_value":"1803.09591v1","created_at":"2026-05-18T00:20:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.09591","created_at":"2026-05-18T00:20:10Z"},{"alias_kind":"pith_short_12","alias_value":"RVDTRCVXZIQP","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RVDTRCVXZIQPTSXT","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RVDTRCVX","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:f9428030f432a77e15d566b579772bb0f73b516978bdec5ed708937bd5ef8a83","target":"graph","created_at":"2026-05-18T00:20:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Chih-Wei Chen, Zhenlei Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-03-26T13:50:38Z","title":"Volume bounds of the Ricci flow on closed manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09591","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:204e8de0b14134acd5f15411bb422239797130f34a30250be8fba08f23d8a1dc","target":"record","created_at":"2026-05-18T00:20:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e1a28ac28f20ddc3e373bb9579256041b0d8b74f323f60221a211800ef32144e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-03-26T13:50:38Z","title_canon_sha256":"6b394740db83bc08d8612fde7c3323abc04c0febacc415a460be3b86e2dd6ea8"},"schema_version":"1.0","source":{"id":"1803.09591","kind":"arxiv","version":1}},"canonical_sha256":"8d47388ab7ca20f9caf38593bcc85cf38428395e6ce624a18337ddf9a9a85969","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8d47388ab7ca20f9caf38593bcc85cf38428395e6ce624a18337ddf9a9a85969","first_computed_at":"2026-05-18T00:20:10.982094Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:10.982094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GrjxLxLyih89drC8ncFNODCVtQCPU4RtOW/ADq5R6kVLrOJB6wXYWqxEAPAOddlS8Ldgb58Ek3UzKznEH5SKAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:10.982848Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.09591","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:204e8de0b14134acd5f15411bb422239797130f34a30250be8fba08f23d8a1dc","sha256:f9428030f432a77e15d566b579772bb0f73b516978bdec5ed708937bd5ef8a83"],"state_sha256":"a17199f986fdad6e180ff1214c36692591c1efda96c0687ab3a082f7c93ffe84"}