{"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"}