Proved
Established result in cited source ↗
Observed Oct 12, 2026
Theorem · version 1
Finite extinction in the simply connected case
Statement
A closed, oriented, simply connected smooth three-manifold with an initial smooth Riemannian metric admits Ricci flow with surgery that becomes extinct in finite time.
Assumptions
- Extinction concerns the flow with surgery, including discarded components; it does not mean that the original manifold vanishes under a homeomorphism.
Evidence
Source reports
Connections
In this problem · publication v13
Poincaré conjecture →
Supporting result · Proved
Relationship not independently reviewed. This concerns its use in this problem.
Sources
A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture →
- §1, Theorems 1–3; §3, release and verification → · Original passage ↗
Recorded wording
The authors report Lean proofs of the smooth theorem and Moise smoothing, yielding the topological theorem. Their release v0.1.3 report names commit 7a48598d35109aa99d1cc678e2724c213cdf4ff3 and audits the endpoint closures for the three standard Lean axioms, without sorryAx or project axioms.
paraphrase · Follow the original source for authoritative statements and reuse terms.
Ziyang Qin · Yuan Liao · Ayush Khaitan · Bennett Chow · Preprint · Publication date Sep 27, 2026
Source-backed selection; publication and proof assessment are separate.
Revision & attribution
Source revision: arxiv-2609.33842v1. Observed Oct 12, 2026.
Authors report automated assistance; OpenMath has not independently assessed that workflow.
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds →
- Abstract; finite-extinction argument → · Original passage ↗
Recorded wording
Perelman establishes finite extinction for the relevant three-manifold class under Ricci flow with surgery; the simply connected closed case is used in the Poincaré proof.
paraphrase · Follow the original source for authoritative statements and reuse terms.
Grigori Perelman · Preprint · Publication date Jul 17, 2003
Source-backed selection; publication and proof assessment are separate.
Revision & attribution
Source revision: arxiv-math-0307245v1. Observed Oct 12, 2026.
AI use is not reported in the selected source.
Ricci Flow and the Poincaré Conjecture →
- Clay monograph 3; introduction, Theorem 0.1 and Corollary 0.2 → · Original passage ↗
Recorded wording
Morgan and Tian give a detailed proof of Perelman’s result, using Ricci flow with surgery and finite-time extinction to obtain the three-sphere conclusion.
paraphrase · Follow the original source for authoritative statements and reuse terms.
John Morgan · Gang Tian · Published · Publication date 2007
Source-backed selection; publication and proof assessment are separate.
Revision & attribution
Source revision: clay-monograph-3-2007. Observed Oct 12, 2026.
AI use is not reported in the selected source.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Established result in cited source ↗
Observed Oct 12, 2026 · Source passage →
Record details
- Claim ID
- claim-poincare-finite-extinction
- Statement version
- 1
- SHA-256
- 14a2ff1cd460de06faeba1f1c9445340abd4fb0110a9592e44f2edf56459b937