The eventually distance minimizing rays in moduli spaces

The eventually distance minimizing ray (EDM ray) in moduli spaces of the Riemann surfaces of analytic finite type with 3g + n − 3 > 0 is studied, which was introduced by Farb and Masur [5]. The asymptotic distance of EDM rays in a moduli space and the distance of end points of EDM rays in the boundary of the moduli space in the augmented moduli space are discussed in this article. A relation between the asymptotic distance of EDM rays and the distance of their end points is established. It is proved also that the distance of end points of two EDM rays is equal to that of end points of two Strebel rays in the Teichmüller space of a covering Riemann surface which are leftings of some representatives of the EDM rays. Meanwhile, simpler proofs for some known results are given.