Weil–Petersson metric
E898484
The Weil–Petersson metric is a natural Kähler metric on Teichmüller space, arising from the \(L^2\)-pairing of quadratic differentials and playing a central role in the geometry of moduli spaces of Riemann surfaces.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Weil–Petersson metric canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991624 — 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: Weil–Petersson metric Context triple: [Teichmüller theory, usesConcept, Weil–Petersson metric]
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A.
Teichmüller curve
A Teichmüller curve is a complex geodesic in the moduli space of Riemann surfaces that arises from flat surface structures and has rich connections to dynamics, geometry, and number theory.
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B.
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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C.
Calabi–Yau metric
A Calabi–Yau metric is a special Ricci-flat Kähler metric with SU(n) holonomy that endows Calabi–Yau manifolds with their characteristic geometric and physical properties.
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D.
Bergman metric
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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E.
Milnor–Wood inequality
The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
- 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: Weil–Petersson metric Target entity description: The Weil–Petersson metric is a natural Kähler metric on Teichmüller space, arising from the \(L^2\)-pairing of quadratic differentials and playing a central role in the geometry of moduli spaces of Riemann surfaces.
-
A.
Teichmüller curve
A Teichmüller curve is a complex geodesic in the moduli space of Riemann surfaces that arises from flat surface structures and has rich connections to dynamics, geometry, and number theory.
-
B.
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.
-
C.
Calabi–Yau metric
A Calabi–Yau metric is a special Ricci-flat Kähler metric with SU(n) holonomy that endows Calabi–Yau manifolds with their characteristic geometric and physical properties.
-
D.
Bergman metric
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.
-
E.
Milnor–Wood inequality
The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Kähler metric
ⓘ
Riemannian metric ⓘ |
| arisesFrom | L^2-pairing of quadratic differentials ⓘ |
| compatibleWith | complex structure on Teichmüller space ⓘ |
| completionContains | noded Riemann surfaces ⓘ |
| completionIs | CAT(0) space ⓘ |
| definedOn |
Teichmüller space
NERFINISHED
ⓘ
moduli space of Riemann surfaces ⓘ |
| definedUsing |
holomorphic quadratic differentials
ⓘ
hyperbolic metrics on Riemann surfaces ⓘ |
| dualSpaceIdentifiedWith | holomorphic quadratic differentials ⓘ |
| extendsTo | completion of Teichmüller space ⓘ |
| hasAssociatedObject |
Weil–Petersson symplectic form
NERFINISHED
ⓘ
Weil–Petersson volume form NERFINISHED ⓘ |
| hasExpressionIn | Fenchel–Nielsen coordinates NERFINISHED ⓘ |
| hasProperty |
Kähler form equals imaginary part of L^2-pairing
ⓘ
Weil–Petersson distance to boundary strata is finite ⓘ Weil–Petersson geodesics may exit Teichmüller space in finite time ⓘ Weil–Petersson volume growth is polynomial in radius on moduli space ⓘ Weil–Petersson volume of moduli space is finite ⓘ curvature bounded above by a negative constant on thick part ⓘ curvature unbounded below near boundary of moduli space ⓘ finite volume on moduli space ⓘ geodesic length functions are real-analytic and strictly convex along Weil–Petersson geodesics ⓘ geodesically convex in thick part of Teichmüller space ⓘ mapping class group acts by isometries ⓘ negative sectional curvature ⓘ variable negative curvature ⓘ |
| induces |
Weil–Petersson distance
NERFINISHED
ⓘ
Weil–Petersson geodesic flow NERFINISHED ⓘ |
| is |
Kähler but not complete
ⓘ
incomplete metric ⓘ not locally symmetric for genus at least 2 ⓘ real-analytic metric ⓘ |
| isInnerProductOn | tangent space of Teichmüller space ⓘ |
| isInvariantUnder | mapping class group NERFINISHED ⓘ |
| isKählerWith | Weil–Petersson symplectic form NERFINISHED ⓘ |
| namedAfter |
André Weil
NERFINISHED
ⓘ
Hans Petersson NERFINISHED ⓘ |
| relatedTo | Fenchel–Nielsen coordinates NERFINISHED ⓘ |
| tangentVectorsCorrespondTo | Beltrami differentials ⓘ |
| usedInStudyOf |
Mirzakhani’s volume recursion for moduli spaces
NERFINISHED
ⓘ
Teichmüller theory NERFINISHED ⓘ asymptotic geometry of moduli space ⓘ geodesic length functions ⓘ geometry of moduli spaces of Riemann surfaces ⓘ hyperbolic surfaces ⓘ mapping class groups ⓘ |
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: Weil–Petersson metric Description of subject: The Weil–Petersson metric is a natural Kähler metric on Teichmüller space, arising from the \(L^2\)-pairing of quadratic differentials and playing a central role in the geometry of moduli spaces of Riemann surfaces.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.