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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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 ⓘ

Referenced by (2)

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 ⓘ