Definition:Teichmüller Annulus

Let $$ R > 0 $$. The set
 * $$A := \C\setminus ([-1,0]\cup [R,\infty)) $$

is called a Teichm&uuml;ller annulus (or also Teichm&uuml;ller extremal domain).

The modulus of $$A$$ is denoted $$\Lambda(R)$$.

Properties
By the Teichm&uuml;ller Modulus Theorem, among all annuli that separate the two points $$0$$ and $$-1$$ both from $$\infty$$ and from a point $$z\in\C$$ with $$|z|=R$$, the Teichm&uuml;ller annulus has the greatest modulus.

The Teichm&uuml;ller annulus is closely related to the Grötzsch annulus.