Teichmüller's theorem

Teichmüller's extremal problem can be stated as follows: given a homeomorphism $ f:X\to X_1$ between Riemann surfaces, find a qc homeomorphism $ f_T:X\to X_1$, homotopic to $ f$, such that $ K(f_T)$ is infimum in this class. We can state the problem in terms of groups as well: given $ \mu \in B(L^\infty({\mathbb{H}},G))$, find $ \mu_T$ with $ \mu_t\sim\mu$ and $ \vert\vert\mu_T\vert\vert _\infty$ infimum in this class. Existence is given by normality properties of quasiconformal mapping (proposition 2.12).

As usual, we assume $ X$ is of type $ (g,n)$ with $ 2g-2+n > 0$. Let $ A_2(X)$ be the space of integrable holomorphic quadratic differentials on $ X$.

Lemma 2.20   For $ \varphi \in A_2(X) \backslash \{0\}$ and $ k \in [0,1)$, set $ \mu_t = \mu(k,\varphi) = k \frac{\overline{\varphi}}{\vert\varphi\vert}$. Then $ \mu_T$ belongs to the unit ball of $ L^\infty_{(-1,1)}(X)$ and $ \vert\mu_T\vert$ is essentially constant on $ X$, with $ \vert\vert\mu_T\vert\vert _\infty = k$. Moreover, $ \mu_T(k,\varphi) = \mu_T(l,\phi)$ if and only if $ k = l$ and $ \varphi = t \phi$, for some $ t > 0$. $ \mu_T$ is called a Teichmüller-Beltrami differential form on $ X$.

Theorem 2.21 (Teichmüller's Theorem)   Given $ [X,f,X_1]$ in $ T(g,n)$, there exists a unique extremal mapping $ f_T$ solving Teichmüller's extremal problem. The complex dilatation of $ f_T$ is a Teichmüller-Beltrami differential for a unique $ k \in (0,1)$ and a unique, up to a positive multiple, element $ \varphi
\in A_2(X)$, if $ [X,f,X_1]$ is not the origin of $ T(g,n)$ (for the origin, $ f_T$ is conformal so $ k=0$).

One can construct explicitly Teichmüller mappings for once-punctured tori as follows. First of all, if $ S_\tau$ is a torus given by $ {\mathbb{C}}/L(1,\tau)$, the group of translations on $ \mathbb{C}$ induces a group of biholomorphic mappings on $ S_\tau$ that acts transitively. So one can assume that $ S_\tau$ has a puncture at (the projection of) the origin. A mapping $ f_\tau:
S_i \to S_\tau$ is given by the affine mapping taking the parallelogram of $ S_i$ (with vertices at 0, $ 1$, $ i$ and $ 1 + i$) to the parallelogram of $ S_\tau$ (with vertices at 0, $ 1$, $ \tau$ and $ 1 + \tau$):

$\displaystyle f_\tau([x+iy] mod L(1,i)) = [x+y\tau] mod L(1,\tau).

To show this mapping is extremal, it is enough to show that $ \mu_{f_\tau}$ is a Teichmüller-Beltrami differential. The lift of $ f_\tau$ to the complex plane is given by $ \widetilde{f_\tau}(z) =
\frac{1}{2} \left( (1-i\tau)z + (1+i\tau)\overline{z} \right)$, so

$\displaystyle \mu_{f_\tau} = \frac{1+i\tau}{1-i\tau} \frac{d\overline{z}}{dz} ...
...}, \hspace{20pt}
\zeta = \frac{1+i\tau}{1-i\tau},\hspace{20pt} \varphi = dz^2,

which shows that $ \mu_{f_\tau}$ is a Teichmüller-Beltrami differential. Teichmüller distance can then be easily computed:

$\displaystyle d_T([X_i,Id,X_i],[X_i,f_\tau,X_\tau]) =
\frac{1}{2}\log K(f_\tau...
...t 1-i\tau\vert + \vert 1+i\tau\vert}{\vert 1-i\tau\vert - \vert 1+i\tau\vert},

which is precisely the Poincaré distance between $ i$ and $ \tau$ in the upper half plane.

Theorem 2.22   Let $ B(A_2(X))$ denote the unit ball of the space of quadratic differentials on $ X$; the mapping $ H_T:B(A_2(X)) \to T(X)$ given by $ H_T(\varphi) = \Phi(\mu_T(\vert\vert\varphi\vert\vert,\varphi)$ is a homeomorphism, called Teichmüller's homeomorphism.

Corollary 2.23   $ T(X)$ ($ X$ a compact surface with punctures and universal covering space the upper half plane) and the Teichmüller metric is simply connected (more than that, it is contractible).

The boundary of $ T(X)$ is Teichmüller embedding is a sphere of dimension $ 6g-7+2n$.

Pablo Ares Gastesi 2005-08-31