"Finite extinction time for the solutions to the Ricci flow on certain three-manifolds"
E684915
"Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" is a landmark mathematical paper by Grigori Perelman that advances the analysis of Ricci flow in three dimensions and plays a key role in his proof of the Poincaré conjecture.
All labels observed (1)
| Label | Occurrences |
|---|---|
| "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7678478 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" Context triple: [Grigori Perelman, notableWork, "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds"]
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A.
Ricci flow
Ricci flow is a geometric evolution equation that smoothly deforms the metric of a Riemannian manifold in a way analogous to heat diffusion, playing a central role in Grigori Perelman's proof of the Poincaré conjecture.
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B.
Kähler–Ricci flow
Kähler–Ricci flow is a geometric evolution equation that deforms Kähler metrics on complex manifolds according to their Ricci curvature, playing a central role in complex differential geometry and the study of canonical metrics.
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C.
Perelman’s entropy functionals
Perelman’s entropy functionals are analytic quantities introduced by Grigori Perelman to study the behavior and singularities of the Ricci flow, playing a central role in his proof of the Poincaré and geometrization conjectures.
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D.
Nirenberg problem in differential geometry
The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
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E.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" Target entity description: "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" is a landmark mathematical paper by Grigori Perelman that advances the analysis of Ricci flow in three dimensions and plays a key role in his proof of the Poincaré conjecture.
-
A.
Ricci flow
Ricci flow is a geometric evolution equation that smoothly deforms the metric of a Riemannian manifold in a way analogous to heat diffusion, playing a central role in Grigori Perelman's proof of the Poincaré conjecture.
-
B.
Kähler–Ricci flow
Kähler–Ricci flow is a geometric evolution equation that deforms Kähler metrics on complex manifolds according to their Ricci curvature, playing a central role in complex differential geometry and the study of canonical metrics.
-
C.
Perelman’s entropy functionals
Perelman’s entropy functionals are analytic quantities introduced by Grigori Perelman to study the behavior and singularities of the Ricci flow, playing a central role in his proof of the Poincaré and geometrization conjectures.
-
D.
Nirenberg problem in differential geometry
The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
-
E.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical paper
ⓘ
research article ⓘ |
| archive | arXiv NERFINISHED ⓘ |
| arXivCategory | math.DG ⓘ |
| associatedConjecture |
Poincaré conjecture
NERFINISHED
ⓘ
Thurston geometrization conjecture NERFINISHED ⓘ |
| author | Grigori Perelman NERFINISHED ⓘ |
| authorFullName | Grigori Yakovlevich Perelman NERFINISHED ⓘ |
| buildsOn | Richard S. Hamilton's work on Ricci flow ⓘ |
| citationForm | Perelman, G. "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds." arXiv preprint math/0307245. ⓘ |
| contributionTo |
proof of the Poincaré conjecture
ⓘ
proof of the geometrization conjecture ⓘ |
| countryOfAuthor | Russia NERFINISHED ⓘ |
| decade | 2000s ⓘ |
| era | early 21st century mathematics ⓘ |
| field |
Riemannian geometry
NERFINISHED
ⓘ
differential geometry ⓘ geometric analysis ⓘ |
| focusesOn |
finite extinction time of Ricci flow solutions
ⓘ
three-dimensional manifolds with certain curvature conditions ⓘ |
| impact | considered a landmark in geometric analysis ⓘ |
| influenced |
research on geometric flows
ⓘ
subsequent work on three-dimensional Ricci flow ⓘ |
| language | English ⓘ |
| mainTopic |
Poincaré conjecture
NERFINISHED
ⓘ
Ricci flow ⓘ geometrization conjecture NERFINISHED ⓘ three-manifolds ⓘ |
| mathematicsSubject |
3-manifold topology
ⓘ
global analysis ⓘ |
| partOfSeries | Perelman's Ricci flow papers NERFINISHED ⓘ |
| proves | finite-time extinction for Ricci flow on certain three-manifolds ⓘ |
| provesPropertyOf | Ricci flow on certain three-manifolds ⓘ |
| publishedAs | arXiv preprint ⓘ |
| relatedWork |
Ricci flow with surgery on three-manifolds
ⓘ
The entropy formula for the Ricci flow and its geometric applications NERFINISHED ⓘ |
| resultType |
existence theorem
ⓘ
extinction time estimate ⓘ |
| status | unpublished in traditional journal at time of release ⓘ |
| usesConcept |
Riemannian metrics
ⓘ
curvature conditions ⓘ |
| usesMethod | Ricci flow with surgery ⓘ |
| year | 2003 ⓘ |
How these facts were elicited
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Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" Description of subject: "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" is a landmark mathematical paper by Grigori Perelman that advances the analysis of Ricci flow in three dimensions and plays a key role in his proof of the Poincaré conjecture.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
Grigori Perelman
→
notableWork
→
"Finite extinction time for the solutions to the Ricci flow on certain three-manifolds"
ⓘ