Definition:Teichmüller Annulus

This page has been identified as a candidate for refactoring of medium complexity. In particular: Put definition of Modulus, in its own page. Before doing that, you need to determine the scope under what conditions the modulus is assigned that denotation. Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

Definition

Let $R \in \R_{>0}$.

The set:

$A := \C \setminus \paren {\closedint{-1} 0 \cup \hointr R {+\infty} }$

is a Teichmüller annulus.

The modulus of $A$ is denoted $\map \Lambda R$.

Also known as

A Teichmüller annulus is also sometimes found referred to as a Teichmüller extremal domain.

Also see

• Teichmüller Modulus Theorem: among all annuli that separate the two points $0$ and $-1$ both from $\infty$ and from a point $z \in \C$ with $\cmod z = R$, the Teichmüller annulus has the greatest modulus.

• Definition:Grötzsch Annulus: a related concept

Source of Name

This entry was named for Oswald Teichmüller.

Sources

• 1973: L.V. Ahlfors: Conformal Invariants: Topics in Geometric Function Theory