TeichmüllerGeodäten

Teichmüller geodesics are curves on a Riemann surface that minimize distance in a specific metric related to the conformal structure of the surface. In the context of Teichmüller space, which parametrizes the conformal structures of a Riemann surface, these geodesics represent straight lines in this abstract space. More formally, they are paths in the Teichmüller space that are geodesics with respect to the Weil-Petersson metric, a natural metric defined on Teichmüller space. The study of Teichmüller geodesics is crucial for understanding the geometry and dynamics of Riemann surfaces. These geodesics can be visualized by considering the associated Bers slices, which are complex analytic subvarieties of the Teichmüller space. The projection of a Teichmüller geodesic onto the surface itself is not necessarily a geodesic in the usual sense of the surface's intrinsic metric. Instead, it relates to how the conformal structure changes along the path. They play a significant role in the theory of dynamical systems on surfaces, particularly in the study of measured geodesic laminations and their connections to extremal quasiconformal maps. The behavior of Teichmüller geodesics, such as their tendency to straighten out as they evolve, is a fundamental aspect of their geometric properties.