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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Nirenberg problem | 0 |
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.