LOWER BOUNDS OF LIPSCHITZ CONSTANTS TO CURVE COMPLEXES OF PUNCTURED RIEMANN SURFACES

摘要

Let S_0 be a closed Riemann surface of genus p > 1.Gadre et al. [4] showed that the optimal Lipschitz constant κ_p for a systol map from the Teíchmüller space T(S_0) to the curve complex C(S_0) behaves like l/log(p) for sufficiently large p. For Riemann surfaces with punctures, Aougab-Taylor showed that there exists a uniform constant L such that κ_p ≥ L/log(3p - 3) for any Riemann surface of type (p, 1) with p > 1. In this paper, we show that k_p≥4/(5log(8p~2 +12p + 3)). As a consequence, we also prove that the constant L can be chosen to satisfy L ≥ 0.4.