Abreu equation
E1017919
equation in Kähler geometry
equation in toric geometry
fourth-order differential equation
nonlinear differential equation
partial differential equation
The Abreu equation is a fourth-order nonlinear partial differential equation arising in Kähler and toric geometry, particularly in the study of extremal and constant scalar curvature Kähler metrics.
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
equation in Kähler geometry
ⓘ
equation in toric geometry ⓘ fourth-order differential equation ⓘ nonlinear differential equation ⓘ partial differential equation ⓘ |
| appearsIn |
constant scalar curvature Kähler (cscK) problem
ⓘ
theory of extremal metrics on toric manifolds ⓘ |
| appliesTo |
Delzant toric varieties
NERFINISHED
ⓘ
compact toric Kähler manifolds ⓘ |
| arisesIn |
Kähler geometry
NERFINISHED
ⓘ
toric geometry ⓘ |
| associatedWith |
Delzant polytopes
NERFINISHED
ⓘ
toric symplectic manifolds ⓘ |
| characterizes |
toric constant scalar curvature Kähler metrics
ⓘ
toric extremal Kähler metrics ⓘ |
| definedOn | moment polytope of a toric Kähler manifold ⓘ |
| domain | convex functions on a Delzant polytope ⓘ |
| field | mathematics ⓘ |
| governs | scalar curvature in symplectic coordinates on toric manifolds ⓘ |
| hasDifferentialOrder | four ⓘ |
| hasOrder | 4 ⓘ |
| introducedBy | Miguel Abreu NERFINISHED ⓘ |
| involves |
Hessian of a convex function
ⓘ
inverse Hessian matrix ⓘ symplectic potential ⓘ |
| isGeometricPDE | true ⓘ |
| isNonlinear | true ⓘ |
| namedAfter | Miguel Abreu NERFINISHED ⓘ |
| relatedTo |
Calabi functional minimization
ⓘ
Monge–Ampère equation NERFINISHED ⓘ extremal vector fields ⓘ |
| relatesTo | scalar curvature of a toric Kähler metric ⓘ |
| requires | boundary conditions on the moment polytope ⓘ |
| studiedIn |
complex differential geometry
ⓘ
geometric analysis ⓘ |
| subfield |
PDE theory
ⓘ
complex geometry ⓘ differential geometry ⓘ symplectic geometry ⓘ |
| usedFor |
study of constant scalar curvature Kähler metrics
ⓘ
study of extremal Kähler metrics ⓘ |
| yearIntroduced | 1998 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.