LENGTH SPECTRUMS OF RIEMANN SURFACES AND THE TEICHMÜLLER METRIC

Abstract

Suppose that $T(S_0)$ is the Teichmüller space of a compact Riemann surface $S_0$ of genus $g>1$. Let $d_T(\cdot,\cdot)$ be the Teichmüller metric of $T(S_0)$ and let $d_S(\cdot,\cdot)$ be a metric of $T(S_0)$ defined by the length spectrums of Riemann surfaces. The author showed in a previous paper that $d_T$ and $d_S$ are topologically equivalent and $d_S(\tau_1,\tau_2)\leq d_T(\tau_1,\tau_2) \leq 2d_S(\tau_1,\tau_2)+C(\tau_1)$, where $C(\tau_1)$ is a constant depending on $\tau_1$. In this paper, it is shown that $d_T$ and $d_S$ are not metrically equivalent; that is, there is no constant $C>0$ such that $d_S(\tau_1,\tau_2)\leq d_T(\tau_1,\tau_2)\leq Cd_S(\tau_1,\tau_2)$ for all $\tau_1 {\rm and} \tau_2 {\rm in} T(S_0)$.