摘要
By explicit constructions,we give direct proofs of the following results： for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g 〉1,there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that lX（α） = lX（β）,extX（α） = extX（β） and lq（α） = lq（β）,where lX（·）,extX（·） and lq（·） are the hyperbolic length,the extremal length and the quadratic differential length respectively.These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.
By explicit constructions,we give direct proofs of the following results： for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g 〉1,there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that lX（α） = lX（β）,extX（α） = extX（β） and lq（α） = lq（β）,where lX（·）,extX（·） and lq（·） are the hyperbolic length,the extremal length and the quadratic differential length respectively.These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.
基金
Supported by NNSF for Young Scientists of China(Grant No.11101290)
NNSF of China(Grant No.11071179)
