Monge–Ampère equation
E326554
elliptic partial differential equation
fully nonlinear equation
mathematical concept
nonlinear elliptic equation
partial differential 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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Monge–Ampère equation canonical | 2 |
| Monge–Ampère equations | 1 |
| Yau’s solution of the Calabi conjecture | 1 |
| complex Monge–Ampère flow | 1 |
| real Monge–Ampère equation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3111321 — 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: Monge–Ampère equation Context triple: [Gaspard Monge, knownFor, Monge–Ampère equation]
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A.
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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B.
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.
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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.
Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
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E.
Singular Integrals and Differentiability Properties of Functions
"Singular Integrals and Differentiability Properties of Functions" is a landmark mathematical monograph by Elias M. Stein that developed the modern theory of singular integral operators and their role in harmonic analysis and differentiability.
- 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: Monge–Ampère equation Target entity description: 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.
-
A.
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.
-
B.
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.
-
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.
Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
-
E.
Singular Integrals and Differentiability Properties of Functions
"Singular Integrals and Differentiability Properties of Functions" is a landmark mathematical monograph by Elias M. Stein that developed the modern theory of singular integral operators and their role in harmonic analysis and differentiability.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
elliptic partial differential equation
ⓘ
fully nonlinear equation ⓘ mathematical concept ⓘ nonlinear elliptic equation ⓘ partial differential equation ⓘ |
| classification | fully nonlinear second-order PDE ⓘ |
| describes |
convex geometry problems
ⓘ
prescribed curvature problems ⓘ |
| field |
differential geometry
ⓘ
optimal transport ⓘ partial differential equations ⓘ several complex variables ⓘ |
| hasForm | det(D^2 u(x)) = f(x,u(x),∇u(x)) ⓘ |
| namedAfter |
André-Marie Ampère
ⓘ
Gaspard Monge ⓘ |
| relatedTo |
Abreu equation
ⓘ
Kantorovich duality ⓘ Monge problem in optimal transport ⓘ
surface form:
Monge–Kantorovich optimal transport problem
curvature of hypersurfaces ⓘ determinant of the Hessian matrix ⓘ real Hessian equations ⓘ |
| requires | convexity of solutions in many real formulations ⓘ |
| solutionType |
classical solution
ⓘ
viscosity solution ⓘ weak solution ⓘ |
| specialCase |
complex Monge–Ampère equation
ⓘ
degenerate Monge–Ampère equation ⓘ Monge–Ampère equation self-linksurface differs ⓘ
surface form:
real Monge–Ampère equation
|
| studiedBy |
Aleksandr Danilovich Aleksandrov
ⓘ
Caffarelli ⓘ Louis Nirenberg ⓘ Shing-Tung Yau ⓘ |
| usedIn |
Christoffel–Minkowski problem
ⓘ
surface form:
Aleksandrov problem
Calabi conjecture ⓘ Kähler geometry ⓘ Christoffel–Minkowski problem ⓘ
surface form:
Minkowski problem
Yau’s proof of the Calabi conjecture ⓘ affine differential geometry ⓘ complex geometry ⓘ construction of Kähler–Einstein metrics ⓘ geometric optics ⓘ image processing ⓘ mass transport problems ⓘ mesh generation ⓘ meteorology ⓘ optimal transport maps ⓘ prescribed Gauss curvature problems ⓘ prescribed Ricci curvature problems ⓘ theory of convex functions ⓘ |
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: Monge–Ampère equation Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Yau’s solution of the Calabi conjecture
this entity surface form:
complex Monge–Ampère flow
this entity surface form:
real Monge–Ampère equation
this entity surface form:
Monge–Ampère equations