Bergman metric
E521039
The Bergman metric is a canonical Kähler metric on complex domains derived from the Bergman kernel, widely used in several complex variables and complex differential geometry.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Bergman metric canonical | 1 |
| the Bergman distance | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5446368 — 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: Bergman metric Context triple: [Carathéodory metric, relatedTo, Bergman metric]
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A.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
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B.
Kähler manifold
A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated symplectic form is closed, making it simultaneously a complex, Riemannian, and symplectic manifold in a compatible way.
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C.
Monge–Ampère 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.
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D.
Kähler form
A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
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E.
Weingarten map
The Weingarten map is a differential geometric operator on a surface that encodes how the surface’s normal vector field changes, thereby describing the surface’s extrinsic curvature.
- 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: Bergman metric Target entity description: The Bergman metric is a canonical Kähler metric on complex domains derived from the Bergman kernel, widely used in several complex variables and complex differential geometry.
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A.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
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B.
Kähler manifold
A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated symplectic form is closed, making it simultaneously a complex, Riemannian, and symplectic manifold in a compatible way.
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C.
Monge–Ampère 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.
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D.
Kähler form
A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
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E.
Weingarten map
The Weingarten map is a differential geometric operator on a surface that encodes how the surface’s normal vector field changes, thereby describing the surface’s extrinsic curvature.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
Hermitian metric
ⓘ
Kähler metric ⓘ canonical metric ⓘ |
| appliesTo |
bounded symmetric domains
ⓘ
pseudoconvex domains ⓘ |
| category | complex Finsler and Riemannian-type metrics ⓘ |
| constructedFrom | Bergman kernel ⓘ |
| curvatureProperty |
has constant holomorphic sectional curvature on the unit ball
ⓘ
has negative holomorphic sectional curvature on the unit ball ⓘ |
| definedOn |
bounded domains in ℂⁿ
ⓘ
complex domains ⓘ |
| definedVia |
Levi form of the logarithm of the Bergman kernel
ⓘ
second derivatives of log K(z,z) where K is the Bergman kernel ⓘ |
| dependsOn | space of square-integrable holomorphic functions ⓘ |
| determines |
an intrinsic complex structure compatible Riemannian metric
ⓘ
the Bergman distance ⓘ
linked to:
Bergman metric
|
| field |
complex differential geometry
ⓘ
several complex variables ⓘ |
| gives | a canonical volume form on the domain ⓘ |
| introducedBy | Stefan Bergman ⓘ |
| introducedIn | 20th century ⓘ |
| invariantUnder |
automorphism group of the domain
ⓘ
biholomorphic maps ⓘ |
| is |
Kähler-Einstein on the unit ball in ℂⁿ
ⓘ
complete on bounded homogeneous domains ⓘ invariant under biholomorphic automorphisms of the domain ⓘ real-analytic on the domain ⓘ unique up to biholomorphic equivalence for a given domain ⓘ |
| regularity | smooth on strongly pseudoconvex domains ⓘ |
| relatedTo |
Carathéodory metric
ⓘ
Kobayashi metric ⓘ Poincaré metric ⓘ |
| restrictionProperty | restricts to the Poincaré metric on the unit disc ⓘ |
| usedFor |
biholomorphic classification of domains
ⓘ
complex Monge–Ampère equations ⓘ defining invariant distances on complex domains ⓘ studying automorphism groups of domains ⓘ studying intrinsic geometry of complex domains ⓘ |
| usedIn |
complex algebraic geometry
ⓘ
complex analysis ⓘ geometric function theory ⓘ theory of several complex variables ⓘ |
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: Bergman metric Description of subject: The Bergman metric is a canonical Kähler metric on complex domains derived from the Bergman kernel, widely used in several complex variables and complex differential geometry.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Bergman metric