EXTREMAL QUASICONFORMAL MAPPINGS WITH PRESCRIBED BOUNDARY VALUES
EXTREMAL QUASICONFORMAL MAPPINGS WITH PRESCRIBED BOUNDARY VALUES
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DOI：
10.1090/s0002-9947-1969-0245787-3
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发表时间：
1969-04
影响因子：
1.3
通讯作者：
R. Hamilton
中科院分区：
文献类型：
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作者：
R. Hamilton
1. 设R为一个黎曼曲面，其普适覆盖空间与单位圆盘保形相等。我们可以把R看作边界为R*的黎曼曲面的内部，它的边界是尽可能大的（见？3）。一个拟共形映射f (R)到另一个黎曼曲面S上，具有唯一的连续扩展f* （R*映射到S*上）。R到S上的两个拟共形映射f和g在R* -R上以边界iff* =g*为同伦模，并且在R*R上f*与g*之间存在一个常数同伦。如果R* R是空的，那么f就是g的同伦。设K是R的复余切束。那么/(f)， f的Beltrami微分，是LO（KK1）的一个元素，LO（KK1）是束RK-1的所有有界部分的Banach空间。（局部v eL-(KK-1)由v dza/dz给出。）L-(KK-1)是L1（K2）的对偶，即束K2的可积截面的巴拿赫空间。（局部L1（K2）由p p dZ给出）设A(R)表示R上所有可积解析二次微分的L1（K2）的闭子空间。通过ll((f) I A(R)II，我们表示/(f)作为L1（K2）上的线性泛函对闭子空间A(R)的约束的范数。拟共形映射f: R -S是极值的当(/3(f) ？（/3(g)）对于所有映射g同伦到f模边界。主要结果是
1. Let R be a Riemann surface whose universal covering space is conformally equivalent to the unit disk. We can regard R as the interior of a Riemann surface with boundary R* whose boundary is as large as possible (see ?3). A quasiconformal map f of R onto another Riemann surface S has a unique continuous extensionf* mapping R* onto S*. Two quasiconformal mapsf and g of R onto S are homotopic modulo the boundary iff* =g* on R* -R and there exists a homotopy between f* and g* which is constant on R*R. If R* R is empty then f is just homotopic to g. Let K be the complex cotangent bundle of R. Then /(f), the Beltrami differential of f, is an element of LO(KK1), the Banach space of all essentially bounded sections of the bundle RK-1. (Locally v eL-(KK-1) is given by v dza/dz,.) L-(KK-1) is the dual of L1(K2), the Banach space of integrable sections of the bundle K2. (Locally e L1(K2) is given by p,P dZ.) Let A(R) denote the closed subspace of L1(K2) of all integrable analytic quadratic differentials on R. By ll((f) I A(R)II we denote the norm of the restriction of /(f), regarded as a linear functional on L1(K2), to the closed subspace A(R). The quasiconformal map f: R -S is extremal if (/3(f) ? (/3(g) ( for all maps g homotopic to f modulo the boundary. The main result is