Teichmüller space
E905423
Teichmüller space is a parameter space in complex analysis and geometry that classifies all marked conformal or hyperbolic structures on a given topological surface up to equivalence.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
moduli space ⓘ parameter space ⓘ |
| classifies |
Riemann surface structures up to equivalence
ⓘ
marked conformal structures on a surface ⓘ marked hyperbolic structures on a surface ⓘ |
| definedFor | oriented topological surface of finite type ⓘ |
| dimensionOverC | 3g-3+n for a surface of genus g with n punctures ⓘ |
| dimensionOverR | 6g-6+2n for a surface of genus g with n punctures ⓘ |
| equivalenceRelation |
conformal equivalence via isotopy of markings
ⓘ
isotopy class of homeomorphisms to a fixed reference surface ⓘ |
| field |
complex analysis
ⓘ
differential geometry ⓘ low-dimensional topology ⓘ |
| generalizationOf | upper half-plane as Teichmüller space of a torus ⓘ |
| hasBoundaryConstruction |
Bers compactification
NERFINISHED
ⓘ
Gardiner–Masur compactification NERFINISHED ⓘ Thurston compactification NERFINISHED ⓘ |
| hasCoordinateDescription |
Bers coordinates
ⓘ
Fenchel–Nielsen coordinates NERFINISHED ⓘ |
| hasMetric |
Teichmüller metric
NERFINISHED
ⓘ
Weil–Petersson metric NERFINISHED ⓘ |
| hasStructure |
Finsler manifold
NERFINISHED
ⓘ
complex manifold ⓘ metric space ⓘ real-analytic manifold ⓘ |
| isConnected | true ⓘ |
| isContractible | true ⓘ |
| isSimplyConnected | true ⓘ |
| isUniversalCoverOf | moduli space of Riemann surfaces NERFINISHED ⓘ |
| namedAfter | Oswald Teichmüller NERFINISHED ⓘ |
| quotientBy | mapping class group NERFINISHED ⓘ |
| quotientGives |
moduli space of Riemann surfaces
NERFINISHED
ⓘ
moduli space of curves ⓘ |
| relatedConcept |
Beltrami differential
NERFINISHED
ⓘ
Bers embedding NERFINISHED ⓘ Fenchel–Nielsen coordinates NERFINISHED ⓘ Weil–Petersson symplectic form NERFINISHED ⓘ mapping class group ⓘ moduli space of algebraic curves NERFINISHED ⓘ quadratic differential ⓘ quasiconformal map ⓘ |
| studiedBy |
Lars Ahlfors
NERFINISHED
ⓘ
Lipman Bers NERFINISHED ⓘ Oswald Teichmüller NERFINISHED ⓘ |
| usedIn |
algebraic geometry
ⓘ
dynamics on moduli spaces ⓘ geometric group theory ⓘ hyperbolic geometry of surfaces ⓘ string theory ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.