On the existence of extremal Teichmueller mappings
On the existence of extremal Teichmueller mappings
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DOI：
10.1007/bf02786734
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发表时间：
1976-12
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通讯作者：
K. Strebel
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文献类型：
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作者：
K. Strebel
设h是单位圆Iz]= 1到Iw [= 1的保向同胚，它有一个到Iz I< 1的拟共形（qc）扩张f.所有这类扩张的类记为F。设子类Fo CF由所有具有给定边界值h的极值qc映射f0组成，即F中具有最小最大伸缩量K 0的映射。Fo永远不会空。我们感兴趣的问题是，如果存在一个Teichmiiller映射在Fo，即映射Fo的复伸缩r 0 = ko（（Oo/I Oo I），其中~ 0 o在I z [< 1（并变换为二次微分）和ko=（K 0 -1）/（Ko+ 1）. Fo中映射的复分解可刻划如下（[3]，[6]）：设r是映射f~ F的复伸缩。则f~ Fo（在这种情况下我们称r是极值）当且仅当存在一列全纯二次微分q~。在I z [< 1，范数为I t 0。II= ftzl < l] o。(z)l dxdy= 1，使得
Let h be an orientation preserving homeomorphism of the unit circle I z]= 1 onto I w [= 1 which has a quasiconformal (qc) extension f into I z I< 1. The class of all such extensions is denoted by F. Let the subclass Fo CF consist of all extremal qc mappings f0 with the given boundary values h, ie mappings in F with smallest maximal dilatation K0. Fo is never empty. We are interested in the problem if there exists a Teichmiiller mapping in Fo, ie a mapping fo with a complex dilatation r0= ko ((Oo/I Oo I), where~ 0o is holomorphic in I z [< 1 (and transforms as a quadratic differential) and ko=(K0-1)/(Ko+ 1).The complex dilatations of the mappings in Fo can be characterized as follows ([3],[6]): Let r be the complex dilatation of a mapping f~ F. Then f~ Fo (in which case we say that r is extremal) if and only if there exists a sequence of holomorphic quadratic differentials q~. in I z [< 1 with norm It o. II= fftzl< l] o.(z) l dxdy= 1 such that