Teichmüller metric
E898485
The Teichmüller metric is a natural Finsler metric on Teichmüller space that measures the minimal quasiconformal distortion needed to deform one Riemann surface into another.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
Finsler metric
ⓘ
intrinsic metric ⓘ metric on Teichmüller space ⓘ |
| appliesTo | Riemann surfaces of finite type ⓘ |
| characterizedBy |
Finsler norm on tangent space of Teichmüller space
ⓘ
extremal quasiconformal maps ⓘ minimal quasiconformal dilatation ⓘ |
| coincidesWith | Kobayashi metric on Teichmüller space ⓘ |
| definedOn |
Teichmüller space
NERFINISHED
ⓘ
space of marked Riemann surfaces ⓘ |
| dependsOn | conformal structures on the surface ⓘ |
| distanceBetweenPointsRepresents | logarithm of minimal quasiconformal dilatation GENERATED ⓘ |
| distanceZeroCondition | two marked Riemann surfaces are equivalent in Teichmüller space GENERATED ⓘ |
| field |
Teichmüller theory
NERFINISHED
ⓘ
complex analysis ⓘ differential geometry ⓘ geometric topology ⓘ |
| generalizes | Poincaré metric on the unit disk (via identification with Teichmüller space of the torus) ⓘ |
| geodesicsAre | Teichmüller geodesics ⓘ |
| geodesicsGivenBy | integrating quadratic differentials ⓘ |
| hasProperty |
non-positively curved in the sense of Teichmüller theory (not CAT(0))
ⓘ
proper metric ⓘ uniquely geodesic in many directions ⓘ |
| introducedBy | Oswald Teichmüller NERFINISHED ⓘ |
| is | the standard metric used in Teichmüller theory ⓘ |
| isAsymmetric | false ⓘ |
| isComplete | true ⓘ |
| isGeodesic | true ⓘ |
| isInvariantUnder |
biholomorphic automorphisms of Teichmüller space
ⓘ
mapping class group action ⓘ |
| isNot | Riemannian metric in general ⓘ |
| isSymmetric | true ⓘ |
| namedAfter | Oswald Teichmüller NERFINISHED ⓘ |
| relatedConcept |
Carathéodory metric
NERFINISHED
ⓘ
Kobayashi metric NERFINISHED ⓘ extremal length ⓘ holomorphic quadratic differential ⓘ quasiconformal mapping ⓘ |
| tangentNormDefinedBy | supremum over unit-area holomorphic quadratic differentials ⓘ |
| topologyInducedEquals | standard topology on Teichmüller space ⓘ |
| usedToStudy |
Teichmüller geodesic flow
NERFINISHED
ⓘ
hyperbolic structures on surfaces ⓘ mapping class group dynamics ⓘ moduli space of Riemann surfaces ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.