TeichmüllerAbbildungen

Teichmüller Abbildungen, also known as Teichmüller mappings, are a type of quasiconformal mapping studied in the field of complex analysis and geometric function theory. They are named after the German mathematician Oswald Teichmüller. A Teichmüller mapping is a homeomorphism between two Jordan curves (or more generally, homeomorphic to a circle) on the Riemann sphere that is absolutely continuous on almost every line and has bounded Dirichlet energy.

More formally, let $f$ be a homeomorphism between two domains $D_1$ and $D_2$ in the complex plane. If $f$ is differentiable almost everywhere and its partial derivatives are in $L^2$, and if its derivative matrix $Df$ satisfies certain distortion properties related to its dilatation, then $f$ is a Teichmüller mapping. The dilatation of a quasiconformal mapping is a measure of how much it distorts angles. Teichmüller mappings are a specific subclass of quasiconformal mappings.

The study of Teichmüller mappings is deeply connected to the Teichmüller space, which is a moduli space of Riemann surfaces. Teichmüller's theorem states that for any two distinct Riemann surfaces of a fixed genus $g > 1$, there exists a unique shortest geodesic in the Teichmüller space connecting their corresponding points. This geodesic is realized by a specific Teichmüller mapping. These mappings play a crucial role in understanding the geometry and topology of Riemann surfaces and their deformations.