Nirenberg problem in differential geometry
E588691
The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Nirenberg problem in differential geometry canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6376277 — 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: Nirenberg problem in differential geometry Context triple: [Louis Nirenberg, knownFor, Nirenberg problem in differential geometry]
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A.
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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B.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
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C.
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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D.
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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E.
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.
- 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: Nirenberg problem in differential geometry Target entity description: The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
-
A.
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.
-
B.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
prescribed curvature problem
ⓘ
problem in differential geometry ⓘ |
| ambientManifold | standard 2-sphere S^2 ⓘ |
| asks | which functions on S^2 occur as Gaussian curvature of a conformal metric ⓘ |
| compatibilityCondition | integral of Gaussian curvature equals 4π on S^2 ⓘ |
| concerns | prescribing Gaussian curvature ⓘ |
| context |
conformal geometry
ⓘ
global analysis on manifolds ⓘ |
| curvatureType | Gaussian curvature ⓘ |
| difficulty |
loss of compactness due to conformal invariance
ⓘ
presence of bubbling phenomena ⓘ |
| dimension | 2 ⓘ |
| domain | 2-sphere ⓘ |
| equationType |
Liouville-type equation
ⓘ
nonlinear elliptic partial differential equation ⓘ |
| field |
differential geometry
ⓘ
geometric analysis ⓘ |
| hasApplications |
construction of metrics with prescribed curvature on surfaces
ⓘ
understanding moduli of conformal metrics on S^2 ⓘ |
| hasGeneralization |
higher-dimensional prescribed scalar curvature problems
ⓘ
prescribing Q-curvature problems ⓘ |
| hasObstructions | Kazdan–Warner identity NERFINISHED ⓘ |
| influenced | development of geometric PDE methods ⓘ |
| involves |
blow-up analysis
ⓘ
critical Sobolev exponent in dimension 2 ⓘ degree theory ⓘ variational methods ⓘ |
| manifoldType | sphere ⓘ |
| metricTransformation | conformal deformation ⓘ |
| metricType | Riemannian metric ⓘ |
| namedAfter | Louis Nirenberg NERFINISHED ⓘ |
| originalFormulation | prescribing Gaussian curvature on S^2 via conformal change of the standard metric ⓘ |
| relatedTo |
Kazdan–Warner problem
NERFINISHED
ⓘ
Yamabe problem ⓘ prescribed scalar curvature problem ⓘ |
| requires | Gauss–Bonnet theorem compatibility ⓘ |
| solutionDependsOn |
integral constraints on the curvature
ⓘ
sign changes of the prescribed curvature function ⓘ symmetry properties of the prescribed curvature function ⓘ |
| status | partially solved with existence and nonexistence results ⓘ |
| studiedSince | 1960s ⓘ |
| transformationGroup | conformal group of the sphere ⓘ |
| typicalMethod |
concentration-compactness principle
ⓘ
minimization of associated energy functional ⓘ moving planes method ⓘ topological degree arguments ⓘ |
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: Nirenberg problem in differential geometry Description of subject: The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.