Calabi conjecture
E888043
The Calabi conjecture is a fundamental result in complex differential geometry, proved by Shing-Tung Yau, which characterizes when a compact Kähler manifold admits a unique Ricci-flat Kähler metric in a given Kähler class.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Calabi conjecture canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10808004 — 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: Calabi conjecture Context triple: [Kähler–Ricci flow, relatedTo, Calabi conjecture]
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A.
Donaldson–Uhlenbeck–Yau theorem
The Donaldson–Uhlenbeck–Yau theorem is a fundamental result in differential and algebraic geometry that characterizes when a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian–Einstein metric, linking geometric stability with the existence of such metrics.
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B.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
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C.
Yamabe problem
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.
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D.
Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
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E.
Strominger–Yau–Zaslow conjecture
The Strominger–Yau–Zaslow conjecture is a proposal in mirror symmetry stating that mirror pairs of Calabi–Yau manifolds can be understood as dual special Lagrangian torus fibrations, providing a geometric explanation of mirror symmetry.
- 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: Calabi conjecture Target entity description: The Calabi conjecture is a fundamental result in complex differential geometry, proved by Shing-Tung Yau, which characterizes when a compact Kähler manifold admits a unique Ricci-flat Kähler metric in a given Kähler class.
-
A.
Donaldson–Uhlenbeck–Yau theorem
The Donaldson–Uhlenbeck–Yau theorem is a fundamental result in differential and algebraic geometry that characterizes when a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian–Einstein metric, linking geometric stability with the existence of such metrics.
-
B.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
C.
Yamabe problem
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.
-
D.
Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
-
E.
Strominger–Yau–Zaslow conjecture
The Strominger–Yau–Zaslow conjecture is a proposal in mirror symmetry stating that mirror pairs of Calabi–Yau manifolds can be understood as dual special Lagrangian torus fibrations, providing a geometric explanation of mirror symmetry.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
result in complex differential geometry ⓘ |
| concerns |
Calabi–Yau manifolds
NERFINISHED
ⓘ
Kähler classes NERFINISHED ⓘ Ricci curvature ⓘ Ricci-flat Kähler metrics ⓘ compact Kähler manifolds ⓘ complex Monge–Ampère equation NERFINISHED ⓘ first Chern class ⓘ |
| dimension | holds in all complex dimensions ⓘ |
| field |
Kähler geometry
NERFINISHED
ⓘ
Riemannian geometry ⓘ algebraic geometry ⓘ complex differential geometry ⓘ |
| formulatedBy | Eugenio Calabi NERFINISHED ⓘ |
| generalizationOf | problems of finding metrics with prescribed Ricci curvature ⓘ |
| hasConsequence |
applications in string theory via Calabi–Yau compactifications
ⓘ
classification of Calabi–Yau manifolds as Ricci-flat Kähler manifolds with vanishing first Chern class ⓘ construction of metrics with prescribed Ricci form ⓘ existence of Kähler–Einstein metrics with zero Ricci curvature ⓘ |
| implies |
existence of Calabi–Yau metrics
ⓘ
existence of Ricci-flat metrics on K3 surfaces ⓘ existence of Ricci-flat metrics on complex tori ⓘ |
| influenced |
development of Calabi–Yau geometry
ⓘ
research in string theory compactifications ⓘ study of Kähler–Einstein metrics ⓘ |
| namedAfter | Eugenio Calabi NERFINISHED ⓘ |
| originallyFormulated | 1950s GENERATED ⓘ |
| provedBy | Shing-Tung Yau NERFINISHED ⓘ |
| provedUsing |
Moser iteration
NERFINISHED
ⓘ
Schauder estimates NERFINISHED ⓘ a priori estimates ⓘ continuity method ⓘ maximum principle ⓘ |
| relatedTo |
Aubin–Yau theorem
NERFINISHED
ⓘ
Calabi–Yau manifold NERFINISHED ⓘ Kähler–Einstein metric NERFINISHED ⓘ Yau's theorem NERFINISHED ⓘ |
| requiresCondition |
compactness of the Kähler manifold
ⓘ
fixed Kähler class ⓘ prescribed first Chern class ⓘ |
| states |
that on a compact Kähler manifold with vanishing first Chern class there exists a Ricci-flat Kähler metric in any given Kähler class
ⓘ
that the Ricci-flat Kähler metric in a fixed Kähler class is unique ⓘ |
| status | proved ⓘ |
| uses |
complex Monge–Ampère equation
NERFINISHED
ⓘ
nonlinear elliptic partial differential equations ⓘ |
| yearProved |
1976
ⓘ
1977 ⓘ |
How these facts were elicited
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Input
Subject: Calabi conjecture Description of subject: The Calabi conjecture is a fundamental result in complex differential geometry, proved by Shing-Tung Yau, which characterizes when a compact Kähler manifold admits a unique Ricci-flat Kähler metric in a given Kähler class.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.