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.
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
NERFINISHED
ⓘ
differential geometry ⓘ geometric analysis ⓘ |
| finallyResolvedBy | Richard Schoen NERFINISHED ⓘ |
| furtherRefinedBy | Thierry Aubin NERFINISHED ⓘ |
| gapCorrectedBy | Neil Trudinger NERFINISHED ⓘ |
| hasGeneralization |
Yamabe-type problems on noncompact manifolds
ⓘ
prescribed scalar curvature problem ⓘ |
| hasHistoricalIssue | gap in Yamabe's original proof ⓘ |
| hasInvariantAssociated |
Yamabe constant
NERFINISHED
ⓘ
sigma invariant of a manifold ⓘ |
| hasVariant |
Yamabe problem in the CR (Cauchy–Riemann) setting
NERFINISHED
ⓘ
Yamabe problem on manifolds with boundary NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| originalWorkBy | Hidehiko Yamabe NERFINISHED ⓘ |
| relatedTo |
Einstein metrics
ⓘ
Sobolev inequalities NERFINISHED ⓘ Yamabe invariant NERFINISHED ⓘ conformal geometry ⓘ critical exponent problems ⓘ |
| solutionCompletedBy |
Neil Trudinger
NERFINISHED
ⓘ
Richard Schoen NERFINISHED ⓘ Thierry Aubin NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
conformal Laplacian NERFINISHED ⓘ |
| yearProposed | 1960 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.