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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (7)

Label	Occurrences
Monge–Ampère equation canonical	4
complex Monge–Ampère equation	3
Monge–Ampère equations	2
Bernstein problem in differential geometry	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]

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
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–Kantorovich optimal transport problem ⓘ linked to: Monge problem in optimal transport 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 ⓘ real Monge–Ampère equation ⓘ linked to: Monge–Ampère equation
studiedBy	Aleksandr Danilovich Aleksandrov ⓘ Caffarelli ⓘ Louis Nirenberg ⓘ Shing-Tung Yau ⓘ
usedIn	Aleksandrov problem ⓘ linked to: Christoffel–Minkowski problem Calabi conjecture ⓘ Kähler geometry ⓘ Minkowski problem ⓘ linked to: Christoffel–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