Cheeger–Gromov compactness theorem
E888038
The Cheeger–Gromov compactness theorem is a fundamental result in Riemannian geometry that gives conditions under which a sequence of Riemannian manifolds has a subsequence converging (in the Gromov–Hausdorff or smooth sense) to a limit space.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cheeger–Gromov compactness theorem canonical | 2 |
| Gromov’s precompactness theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10807890 — 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: Cheeger–Gromov compactness theorem Context triple: [Hamilton’s compactness theorem, relatedTo, Cheeger–Gromov compactness theorem]
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A.
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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B.
Rellich–Kondrachov compactness theorem
The Rellich–Kondrachov compactness theorem is a fundamental result in functional analysis and the theory of Sobolev spaces that guarantees the compactness of certain embedding operators, playing a key role in the study of partial differential equations.
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C.
Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
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D.
Bochner technique in Riemannian geometry
The Bochner technique in Riemannian geometry is a method that uses Bochner-type identities and curvature conditions to derive vanishing theorems and rigidity results for differential forms and harmonic maps on manifolds.
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E.
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.
- 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: Cheeger–Gromov compactness theorem Target entity description: The Cheeger–Gromov compactness theorem is a fundamental result in Riemannian geometry that gives conditions under which a sequence of Riemannian manifolds has a subsequence converging (in the Gromov–Hausdorff or smooth sense) to a limit space.
-
A.
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.
-
B.
Rellich–Kondrachov compactness theorem
The Rellich–Kondrachov compactness theorem is a fundamental result in functional analysis and the theory of Sobolev spaces that guarantees the compactness of certain embedding operators, playing a key role in the study of partial differential equations.
-
C.
Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
-
D.
Bochner technique in Riemannian geometry
The Bochner technique in Riemannian geometry is a method that uses Bochner-type identities and curvature conditions to derive vanishing theorems and rigidity results for differential forms and harmonic maps on manifolds.
-
E.
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.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
compactness theorem
ⓘ
mathematical theorem ⓘ |
| appliesTo |
compact Riemannian manifolds
ⓘ
families of manifolds with bounded geometry ⓘ pointed complete Riemannian manifolds ⓘ |
| assumes |
uniform bounds on curvature and its derivatives in C^k or C^∞ versions
ⓘ
uniform lower bound on Ricci curvature or sectional curvature ⓘ uniform lower bound on injectivity radius in smooth versions ⓘ uniform upper bound on diameter in many formulations ⓘ |
| concerns |
Gromov–Hausdorff convergence
ⓘ
precompactness in appropriate topology ⓘ sequences of Riemannian manifolds ⓘ smooth convergence of Riemannian manifolds ⓘ |
| field |
Riemannian geometry
ⓘ
differential geometry ⓘ metric geometry ⓘ |
| givesConditionFor |
Gromov–Hausdorff precompactness
ⓘ
existence of convergent subsequence of Riemannian manifolds ⓘ smooth Cheeger–Gromov convergence ⓘ |
| guarantees | subsequence convergence under uniform geometric bounds ⓘ |
| implies |
existence of convergent subsequence of manifolds under curvature and diameter bounds
ⓘ
limit is a Riemannian manifold in smooth convergence setting ⓘ limit is a compact metric space in Gromov–Hausdorff setting ⓘ |
| limitObject |
Riemannian manifold in smooth Cheeger–Gromov sense
ⓘ
compact metric space in Gromov–Hausdorff sense ⓘ |
| namedAfter |
Jeff Cheeger
ⓘ
Mikhael Gromov ⓘ
linked to:
Mikhail Gromov
|
| relatedTo |
Arzelà–Ascoli theorem
ⓘ
Gromov compactness theorem ⓘ precompactness of isometry classes of manifolds ⓘ |
| typicalHypothesis |
fixed dimension of the manifolds
ⓘ
uniform bound on absolute value of sectional curvature ⓘ uniform positive lower bound on injectivity radius ⓘ |
| usedIn |
Ricci flow theory
ⓘ
linked to:
Ricci flow
geometric analysis ⓘ global Riemannian geometry ⓘ study of collapsing Riemannian manifolds ⓘ study of moduli spaces of Riemannian metrics ⓘ |
| usesConcept |
C^k convergence of Riemannian metrics
ⓘ
Gromov–Hausdorff distance ⓘ Ricci curvature ⓘ
linked to:
Ricci curvature tensor
curvature tensor ⓘ injectivity radius ⓘ pointed Gromov–Hausdorff convergence ⓘ sectional curvature ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
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: Cheeger–Gromov compactness theorem Description of subject: The Cheeger–Gromov compactness theorem is a fundamental result in Riemannian geometry that gives conditions under which a sequence of Riemannian manifolds has a subsequence converging (in the Gromov–Hausdorff or smooth sense) to a limit space.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Cheeger–Gromov compactness theorem