Teichmüller curve
E262445
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Teichmüller curve canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2408414 — 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: Teichmüller curve Context triple: [Klein quartic, moduliSpacePoint, Teichmüller curve]
-
A.
Teichmüller theory
Teichmüller theory is a branch of complex analysis and geometry that studies the deformation spaces of Riemann surfaces and their moduli, often via quasiconformal mappings.
-
B.
Riemann surfaces
Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.
-
C.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
D.
uniformization theorem
The uniformization theorem is a fundamental result in complex analysis stating that every simply connected Riemann surface is conformally equivalent to either the Riemann sphere, the complex plane, or the unit disk.
-
E.
Kleinian group
A Kleinian group is a discrete subgroup of Möbius transformations acting on hyperbolic 3-space, central to the study of Riemann surfaces, complex dynamics, and low-dimensional topology.
- 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: Teichmüller curve Target entity description: 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.
-
A.
Teichmüller theory
Teichmüller theory is a branch of complex analysis and geometry that studies the deformation spaces of Riemann surfaces and their moduli, often via quasiconformal mappings.
-
B.
Riemann surfaces
Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.
-
C.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
D.
uniformization theorem
The uniformization theorem is a fundamental result in complex analysis stating that every simply connected Riemann surface is conformally equivalent to either the Riemann sphere, the complex plane, or the unit disk.
-
E.
Kleinian group
A Kleinian group is a discrete subgroup of Möbius transformations acting on hyperbolic 3-space, central to the study of Riemann surfaces, complex dynamics, and low-dimensional topology.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic curve
ⓘ
complex geodesic ⓘ geodesic in moduli space ⓘ mathematical object ⓘ |
| arisesFrom |
Veech surface with lattice Veech group
ⓘ
flat surface ⓘ half-translation surface ⓘ pair (X,ω) of Riemann surface and holomorphic 1-form ⓘ quadratic differential on a Riemann surface ⓘ translation surface ⓘ |
| constructedBy | SL(2,R)-orbit of a flat surface in a stratum of differentials ⓘ |
| definedIn |
Teichmüller space
ⓘ
moduli space of Riemann surfaces ⓘ |
| field |
Teichmüller theory
ⓘ
complex analysis ⓘ differential geometry ⓘ dynamical systems ⓘ number theory ⓘ |
| hasMonodromy |
Fuchsian group
ⓘ
surface form:
Veech group
|
| hasProperty |
Kobayashi geodesic in moduli space
ⓘ
algebraic curve in the moduli space of curves ⓘ can be Teichmüller–Shimura curves in special cases ⓘ complex geodesic for the Teichmüller metric ⓘ finite area quotient of the hyperbolic plane ⓘ geodesic for the Weil–Petersson metric only in special cases ⓘ isometrically immersed for the Teichmüller metric ⓘ often defined over number fields ⓘ often has arithmetic significance ⓘ often has special Jacobians with extra endomorphisms ⓘ projection of a Teichmüller disk to moduli space ⓘ stabilizer in SL(2,R) is a lattice ⓘ totally geodesic for the Teichmüller metric ⓘ |
| hasStructure | complex one-dimensional submanifold of moduli space ⓘ |
| imageOf | Teichmüller disk with discrete stabilizer ⓘ |
| namedAfter | Oswald Teichmüller ⓘ |
| relatedTo |
Kontsevich–Zorich cocycle
ⓘ
Lyapunov exponents of the Hodge bundle ⓘ Shimura varieties ⓘ
surface form:
Shimura curve
Veech surface ⓘ billiards in rational polygons ⓘ dynamics of straight-line flows on translation surfaces ⓘ interval exchange transformations ⓘ moduli space of Abelian differentials ⓘ moduli space of quadratic differentials ⓘ |
| studiedBy |
Anton Zorich
ⓘ
Curtis T. McMullen ⓘ
surface form:
Curtis McMullen
Howard Masur ⓘ Martin Möller ⓘ |
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: Teichmüller curve Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.