nLab Teichmüller theory

Idea

The Teichmüller space $\mathcal{T}_{\Sigma}$ of a (closed) 2-dimensional manifold $\Sigma$ is the moduli space of complex structures on $\Sigma$, where two complex structures are identified if they are taken into each other by a homeomorphism $\phi \colon \Sigma \to \Sigma$ which is isotopic to the identity.

Since the diffeomorphism class of a closed manifold of dimension 2 is given by its genus $g\in \mathbb{Z}$, typically one speaks of the Teichmüller space $\mathcal{T}_g$ for each $g$. More generally one considers $\Sigma$s equipped with $n$ punctures/boundary circle and then writes $\mathcal{T}_{g,n}$.

Teichmüller space is a covering space for the moduli space of curves over the complex numbers (see below). When also the singular nodal curve is included one speaks of the augmented Teichmüller space which in turn is a cover of the Deligne-Mumford compactification of the moduli space of complex curves.

Properties

Complex structure on Teichmüller space

Teichmüller space itself carries a complex structure.

This was envisioned in (Teichmüller 44) and proven in (Ahlfors 60, Bers 60).

Review includes (Schumacher, section 2)

Relation to moduli stack of complex curves / Riemann surfaces

Quotienting the space of complex structures on $\Sigma$ by all homeomorphisms produces what is called the moduli space of curves (over the complex numbers). Closely related to the moduli space of conformal structures hence of Riemann surfaces.

Hence the Riemann moduli space is the orbifold quotient of Teichmüller space by the mapping class group of $\Sigma$. See (Hubbard-Koch 13).

Related concepts

• Kodaira-Spencer theory

• moduli space of curves

• Grothendieck-Teichmüller group

• quantum Teichmüller theory

• p-adic Teichmüller theory

• inter-universal Teichmüller theory

• Outer space

• for version in supergeometry see at super Riemann surface

References

The original articles are

• Oswald Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Preußische Akademie der Wissenschaften, nat. Kl. 22 1-197 (1939)

• Oswald Teichmüller, Veränderliche Riemannsche Flächen, Deutsche Math. 7 344-359 (1944). English translation: Variable Riemann surfaces, In: Athanase Papadopoulos. Handbook of Teichmüller Theory, Volume IV, 19, European Mathematical Society, pp 787–803, 2014, IRMA Lectures in Mathematics and Theoretical Physics, doi:10.4171/117-1/19.

and see also

• Annette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji, Athanase Papadopoulos. A commentary on Teichmüller’s paper “Veränderliche Riemannsche Flächen” (Variable Riemann Surfaces), In: Athanase Papadopoulos. Handbook of Teichmüller Theory, Volume IV, 19, European Mathematical Society, pp.804-814, 2014, IRMA Lectures in Mathematics and Theoretical Physics, doi:10.4171/117-1/20, hal-00730238, arXiv:1209.1882.

The complex structure on Teichmüller spaces was fully established in

• L. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces, in Analytic Functions, Princeton University Press (1960)

• Lipman Bers, Spaces of Riemann surfaces, Proc. Int. Cong. of Mathematics 1958, Cambridge 1960

Reviews include

• Georg Schumacher, The theory of Teichmüller spaces – A view towards moduli spaces of Kähler manifolds (pdf)

• Wikipedia, Teichmüller space

Further developments include

• Alexander Grothendieck, Techniques de construction en géométrie analytique. I. Description axiomatique de l’espace de Teichmüller et de ses variantes Séminaire Henri Cartan 13 no. 1, Exposés No. 7 and 8 (Paris: Secrétariat Mathématique). (1960/1961) (Numdam)

• John Hubbard, Sarah Koch, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves (arXiv:1301.0062)

Relation to anti-de Sitter spacetime:

• Francesco Bonsante, Andrea Seppi, Anti-de Sitter geometry and Teichmüller theory (arXiv:2004.14414)