Yamabe problem
E603721
The Yamabe problem is a fundamental question in differential geometry concerning whether every compact Riemannian manifold admits a metric of constant scalar curvature within a given conformal class.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Yamabe problem canonical | 2 |
| CR Yamabe problem | 1 |
| Kazdan–Warner problem | 1 |
| Yamabe problem in the CR (Cauchy–Riemann) setting | 1 |
| Yamabe problem on manifolds with boundary | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6514673 — 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: Yamabe problem Context triple: [Richard Schoen, notableWork, Yamabe problem]
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Dirichlet problem
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
- 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: Yamabe problem Target entity description: The Yamabe problem is a fundamental question in differential geometry concerning whether every compact Riemannian manifold admits a metric of constant scalar curvature within a given conformal class.
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Dirichlet problem
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical problem
ⓘ
problem in differential geometry ⓘ |
| alsoKnownAs | prescribed scalar curvature problem in a conformal class ⓘ |
| appliesTo | compact manifolds without boundary ⓘ |
| asksWhether | every compact Riemannian manifold admits a metric of constant scalar curvature in a given conformal class ⓘ |
| concerns |
compact Riemannian manifolds
ⓘ
conformal classes of Riemannian metrics ⓘ metrics of constant scalar curvature ⓘ |
| connectedTo | positive mass theorem (through Schoen's work) ⓘ |
| difficulty | critical nonlinearity of the associated PDE ⓘ |
| dimensionRestriction | manifolds of dimension at least 3 ⓘ |
| field |
Riemannian geometry
ⓘ
linked to:
Riemannian manifolds
differential geometry ⓘ geometric analysis ⓘ |
| finallyResolvedBy | Richard Schoen ⓘ |
| furtherRefinedBy | Thierry Aubin ⓘ |
| gapCorrectedBy | Neil Trudinger ⓘ |
| hasGeneralization |
Yamabe-type problems on noncompact manifolds
ⓘ
prescribed scalar curvature problem ⓘ |
| hasHistoricalIssue | gap in Yamabe's original proof ⓘ |
| hasInvariantAssociated |
Yamabe constant
ⓘ
linked to:
Yamabe invariant
sigma invariant of a manifold ⓘ |
| hasVariant |
Yamabe problem in the CR (Cauchy–Riemann) setting
ⓘ
linked to:
Yamabe problem
Yamabe problem on manifolds with boundary ⓘ
linked to:
Yamabe problem
|
| influenced | development of geometric analysis ⓘ |
| influences | study of Einstein metrics via conformal deformation ⓘ |
| involves |
conformal deformation of metrics
ⓘ
nonlinear elliptic partial differential equations ⓘ scalar curvature ⓘ variational methods ⓘ |
| isSpecialCaseOf | prescribed curvature problems in geometry ⓘ |
| motivated | study of conformal invariants of manifolds ⓘ |
| namedAfter | Hidehiko Yamabe ⓘ |
| originalWorkBy | Hidehiko Yamabe ⓘ |
| relatedTo |
Einstein metrics
ⓘ
Sobolev inequalities ⓘ
linked to:
Sobolev inequality
Yamabe invariant ⓘ conformal geometry ⓘ critical exponent problems ⓘ |
| solutionCompletedBy |
Neil Trudinger
ⓘ
Richard Schoen ⓘ Thierry Aubin ⓘ |
| solutionMethod | minimization of the normalized total scalar curvature functional ⓘ |
| status | solved ⓘ |
| typicalEquationForm | L_g u = c u^{ rac{n+2}{n-2}} on an n-dimensional manifold ⓘ |
| uses |
Sobolev embedding theorem
ⓘ
linked to:
Sobolev inequality
conformal Laplacian ⓘ |
| yearProposed | 1960 ⓘ |
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: Yamabe problem Description of subject: The Yamabe problem is a fundamental question in differential geometry concerning whether every compact Riemannian manifold admits a metric of constant scalar curvature within a given conformal class.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Yamabe problem
subject linked to:
Nirenberg problem in differential geometry
linked to: Yamabe problem
linked to: Yamabe problem
linked to: Yamabe problem