A Note on Strebel Differentials

Abstract

Let S be a hyperbolic Riemann surface and let T(S) be the Teichmuller space of S, viewed as Teichmuller equivalence classes of Beltrami differentials. The following theorem is proved in this paper: Suppose τ 0 is an arbitrarily given point of T(S) and τ is a Strebel point of T(S). Then for any extremal Beltrami differential μ of τ 0 we have $$ 0 \leqslant \left\| {\left. \mu \right\|} \right.-Re\iint_s {\mu (z){\phi _\tau }(z)dsdy \leqslant C\left[ {exp({d_T}({\tau _0},\tau ))-1} \right],} $$ where d T is the Teichmuller distance and C is a constant. As a consequence of this inequality, for any point τ 0 in T(S), there is a Hamilton sequence φ n for any extremal differential of τ 0 formed by Strebel differentials.