On Equivalence of Simple Closed Curves in Flat Surfaces  

作　　者：Zong Liang SUN 

机构地区：[1]Department of Mathematics,Shenzhen University

出　　处：《Acta Mathematica Sinica,English Series》2014年第11期1827-1832,共6页数学学报（英文版）

基　　金：Supported by NNSF for Young Scientists of China(Grant No.11101290);NNSF of China(Grant No.11071179)

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.

关 键 词：Hyperbolic metric quadratic differential metric simple closed curve 

分 类 号：O186.11[理学—数学]