Kobayashi’s and Teichmuller’s Metrics and Bers Complex Manifold Structure on Circle Diffeomorphisms  

作　　者：Yun Ping JIANG 

机构地区：[1]Department of Mathematics,Queens College of the City University of New York Flushing,NY 11367-1597,USA [2]Department of Mathematics,Graduate School of the City University of New York,365 Fifth A venue,New York,NY 10016,USA

出　　处：《Acta Mathematica Sinica,English Series》2020年第3期245-272,共28页数学学报（英文版）

基　　金：supported by the National Science Foundation;supported by a collaboration grant from the Simons Foundation(Grant No.523341);PSC-CUNY awards and a grant from NSFC(Grant No.11571122)。

摘　　要：Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk.We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms.Using these distortion properties,we give the Bers complex manifold structure on the Teichm(u| ")ller space TC^1+H as the union of over all0 <α≤1,which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure.Furthermore,we prove that with the Bers complex manifold structure on TC^1+H ,Kobayashi’s metric and Teichmuller’s metric coincide.Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk.We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms.Using these distortion properties,we give the Bers complex manifold structure on the Teichm（u| "）ller space TC1+H as the union of over all0 <α≤1,which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure.Furthermore,we prove that with the Bers complex manifold structure on TC1+H,Kobayashi’s metric and Teichmuller’s metric coincide.

关 键 词：Bers complex manifold STRUCTURE circle DIFFEOMORPHISM modulus of continuity quasisymmetric circle HOMEOMORPHISM Teichmuller space Kobayashi's METRIC Teichmuller's METRIC 

分 类 号：O189.31[理学—数学]