Weil–Petersson metric

E898484

Kähler metric Riemannian metric

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.

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All labels observed (4)

Label	Occurrences
Weil–Petersson symplectic form	4
Weil–Petersson metric canonical	3
Weil–Petersson distance	1
Weil–Petersson geometry	1

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]

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
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 ⓘ 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 ⓘ linked to: Weil–Petersson metric Weil–Petersson volume form ⓘ
hasExpressionIn	Fenchel–Nielsen coordinates ⓘ
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 ⓘ linked to: Weil–Petersson metric Weil–Petersson geodesic flow ⓘ
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 ⓘ
isKählerWith	Weil–Petersson symplectic form ⓘ linked to: Weil–Petersson metric
namedAfter	André Weil ⓘ Hans Petersson ⓘ
relatedTo	Fenchel–Nielsen coordinates ⓘ
tangentVectorsCorrespondTo	Beltrami differentials ⓘ
usedInStudyOf	Mirzakhani’s volume recursion for moduli spaces ⓘ linked to: moduli space of Riemann surfaces Teichmüller theory ⓘ 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

Referenced by (9)

Full triples — surface form annotated when it differs from this entity's canonical label.

Teichmüller theory → usesConcept → Weil–Petersson metric ⓘ

Teichmüller theory → hasMetricStructure → Weil–Petersson metric ⓘ

Jeffrey Brock → hasResearchInterest → Weil–Petersson geometry ⓘ