Some notes on Teichmüller shift mappings and the Teichmüller density

Abstract

Kra [Kr] introduced a distance $d_{K}$ on every hyperbolic Riemann surface $R$ by means of Teichmüller shift mappings. Recently Gardiner and Lakic [GL2] defined a metric density $\lambda$, the Teichmüller density, on such a surface. The paper deals with some basic properties of the Teichmüller density $\lambda$ and the distance $d_{K}$, giving some close relation between them. Particularly, it is shown that the distance function $d_{K} : R \times R \to \mathbb{R}$ is continuously differentiable off the diagonal, and the Teichmüller density $\lambda$ is precisely the metric density of the infinitesimal form of the distance $d_{K}$ and it is continuous on the whole surface $R$. Some related topics will also be discussed.