Question

I'm having a brain freeze.

Let $B$ be the space of complex valued measurable functions on the unit disk in the complex plane with essential supremum less than 1. Then, the universal Teichmuller space $T$ can be though of as a quotient space of $B$.

I vaguely recall reading somewhere that for every point $p \in T$ has a unique representative of the form

$\mu(z) = k \frac{\bar{q}}{q}$, where $q$ is a holomorphic function on the unit disk.

First of all, am I crazy and am remembering something that's completely off?

Secondly, if there is a statement like that, where should I be looking?

Answer 1

1. Strebel discovered (in 1962) that Teichmüller's existence theorem fails for quasiconformal maps of the unit disk. Uniqueness fails too for points in $T(D^2)$ which do not have Teichmuller representatives. See Strebel's examples in the book by Gardiner and Lakic, "Quasiconformal Teichmüller Theory," p. 177-178.

2. On the other hand, the uniqueness theorem in certain sense does hold (Theorem 5, Chapter 4 of the same book). More precisely, if a quasiconformal map $f: D^2\to D^2$ is a Teichmüller map, i.e., it has Beltrami differential of the Teichmüller form $t \bar{\phi}/|\phi|$ (where $\phi$ is holomorphic), then $f$ is the unique extremal quasiconformal map in its equivalence class $[f]\in T(D^2)$. In particular, $[f]$ contains no other Teichmüller maps.

3. On third hand, Strebel proved in 1976 ("On the existence of extremal Teichmueller mappings") that for $[f]\in T(D^2)$ given by, say, smooth boundary values, the Teichmüller existence theorem does hold.

Answer 2

I think you probably referred to this fact: you can find for your point $p \in T$ a representative in the form of a Beltrami differential $\mu$ expressed as $\mu=k\frac{\bar{\phi}}{\vert \phi \vert}$, with $\phi$ a quadratic differential. But I doubt about the uniqueness, as you can always multiply $\phi$ by some positive constant...

Hubbard's book on Teichm\üller theory contains certainly all of that. Also, I think that Lehto's book has an entire chapter on Universal Teichm\üller space.