MODIFIED LENGTH SPECTRUM METRIC ON THE TEICHMÜLLER SPACE OF A RIEMANN SURFACE WITH BOUNDARY

摘要

Let S0 be a bordered Riemann surface of finite type, and letT(S0) (resp. TR(S0)) be the Teichmüller space (resp.reduced Teichmüller space) of S0. The length spectrum function defines a metric on TR(S0)) but not on T(S0). Inthis paper, we introduce a modified length spectrum function thatdoes define a metric on T(S0). Then we show that if two pointsof T(S0) are close in the Teichmüller metric then they areclose in the modified length spectrum metric, but the converse isnot true. We also prove that T(S0) is not complete under thismodified length spectrum metric.